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authorSebastiano Tronto <sebastiano.tronto@gmail.com>2021-05-25 17:10:49 +0200
committerSebastiano Tronto <sebastiano.tronto@gmail.com>2021-05-25 17:10:49 +0200
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tree118ff3c2424e735149c145524965a4a337e50beb /src/Lecture7/slides
parent46eef66b1e1571c77dc828d7e950b129b4c8bfd0 (diff)
downloadmathsoftware-d6c61d988bfa4255baf9cdae42db59ebee38363f.tar.gz
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1{
2 "cells": [
3 {
4 "cell_type": "markdown",
5 "metadata": {},
6 "source": [
7 "# Nested loops\n",
8 "\n",
9 "The following two functions compute sum and product of matrices, respectively.\n",
10 "\n",
11 "By counting the nested loops it is easy to see that `add()` is $O(n^2)$ while `prod()` is $O(n^3)$."
12 ]
13 },
14 {
15 "cell_type": "code",
16 "execution_count": 37,
17 "metadata": {},
18 "outputs": [
19 {
20 "name": "stdout",
21 "output_type": "stream",
22 "text": [
23 "Time for add: 0.00012074300000008975\n",
24 "Time for prod: 0.00036587199999971176\n"
25 ]
26 }
27 ],
28 "source": [
29 "from random import randint\n",
30 "import time\n",
31 "\n",
32 "def add(A, B):\n",
33 " S = [[0] * len(A) for i in range(len(A))]\n",
34 " for i in range(len(A)):\n",
35 " for j in range(len(A)):\n",
36 " S[i][j] = A[i][j] + B[i][j]\n",
37 " return S\n",
38 "\n",
39 "def prod(A, B):\n",
40 " S = [[0] * len(A) for i in range(len(A))]\n",
41 " for i in range(len(A)):\n",
42 " for j in range(len(A)):\n",
43 " for k in range(len(A)):\n",
44 " S[i][j] = S[i][j] + A[i][k] * B[k][j]\n",
45 " return S\n",
46 "\n",
47 "N = 10\n",
48 "A = [ [randint(0,100) for i in range(N)] for j in range(N) ]\n",
49 "B = [ [randint(0,100) for i in range(N)] for j in range(N) ]\n",
50 "\n",
51 "t0 = time.process_time()\n",
52 "add(A,B)\n",
53 "t1 = time.process_time()\n",
54 "prod(A,B)\n",
55 "t2 = time.process_time()\n",
56 "\n",
57 "print(\"Time for add: \", t1-t0)\n",
58 "print(\"Time for prod:\", t2-t1)"
59 ]
60 },
61 {
62 "cell_type": "markdown",
63 "metadata": {},
64 "source": [
65 "# Sorting a list, slow version\n",
66 "\n",
67 "The following code implements a slow version of the so-called *insertion sort* alogithm\n",
68 "\n",
69 "Complexity: $O(n^2)$."
70 ]
71 },
72 {
73 "cell_type": "code",
74 "execution_count": 61,
75 "metadata": {},
76 "outputs": [
77 {
78 "name": "stdout",
79 "output_type": "stream",
80 "text": [
81 "Running time: 1.1012288430000012\n"
82 ]
83 }
84 ],
85 "source": [
86 "from random import randint\n",
87 "import time\n",
88 "\n",
89 "def correct_position(e, S):\n",
90 " for i in range(len(S)):\n",
91 " if S[i] > e:\n",
92 " return i\n",
93 " return len(S)\n",
94 "\n",
95 "def sort_list(L):\n",
96 " S = []\n",
97 " for e in L:\n",
98 " cp = correct_position(e, S)\n",
99 " S.insert(cp, e)\n",
100 " return S\n",
101 "\n",
102 "N = 10000\n",
103 "L = [randint(0,10**9) for i in range(N)]\n",
104 "\n",
105 "t0 = time.process_time()\n",
106 "sort_list(L)\n",
107 "t1 = time.process_time()\n",
108 "\n",
109 "print(\"Running time:\", t1-t0)"
110 ]
111 },
112 {
113 "cell_type": "markdown",
114 "metadata": {},
115 "source": [
116 "# Binary search\n",
117 "\n",
118 "The following code implements a binary search.\n",
119 "\n",
120 "Complexity: $O(\\log_2(n))$"
121 ]
122 },
123 {
124 "cell_type": "code",
125 "execution_count": 53,
126 "metadata": {},
127 "outputs": [
128 {
129 "name": "stdout",
130 "output_type": "stream",
131 "text": [
132 "The correct position of e = 658230309 in L is:\n",
133 "... 658211821 658224379 e 658234625 658246765 ...\n",
134 "\n",
135 "Time for sorting: 0.021211020999999164\n",
136 "Time for searching: 7.820199999741817e-05\n"
137 ]
138 }
139 ],
140 "source": [
141 "from random import randint\n",
142 "import time\n",
143 "\n",
144 "def binary_search(e, S, start, end):\n",
145 " if start == end:\n",
146 " return start\n",
147 " midpoint = (start+end) // 2\n",
148 " if e < S[midpoint]:\n",
149 " return binary_search(e, S, start, midpoint)\n",
150 " else:\n",
151 " return binary_search(e, S, midpoint+1, end)\n",
152 " \n",
153 "N = 100000\n",
154 "L = [randint(0,10**9) for i in range(N)]\n",
155 "e = randint(0,10**9)\n",
156 "\n",
157 "t0 = time.process_time()\n",
158 "L.sort() # Using Python's sort()\n",
159 "t1 = time.process_time()\n",
160 "i = binary_search(e, L, 0, len(L))\n",
161 "t2 = time.process_time()\n",
162 "print(\"The correct position of e =\", e, \"in L is:\")\n",
163 "print(\"...\", L[i-2], L[i-1], \"e\", L[i], L[i+1], \"...\")\n",
164 "print(\"\")\n",
165 "print(\"Time for sorting: \", t1-t0)\n",
166 "print(\"Time for searching:\", t2-t1)\n"
167 ]
168 },
169 {
170 "cell_type": "markdown",
171 "metadata": {},
172 "source": [
173 "# Sorting a list, fast version (with binary_search)\n",
174 "\n",
175 "The following code uses the function `binary_search()` above instead of `correct_position()` in our insertion sort algorithm.\n",
176 "\n",
177 "Complexity: $O(n\\log_2(n))$"
178 ]
179 },
180 {
181 "cell_type": "code",
182 "execution_count": 69,
183 "metadata": {},
184 "outputs": [
185 {
186 "name": "stdout",
187 "output_type": "stream",
188 "text": [
189 "Running time: 0.03710268399998995\n"
190 ]
191 }
192 ],
193 "source": [
194 "from random import randint\n",
195 "import time\n",
196 "\n",
197 "def binary_search(e, S, start, end):\n",
198 " if start == end:\n",
199 " return start\n",
200 " midpoint = (start+end) // 2\n",
201 " if e < S[midpoint]:\n",
202 " return binary_search(e, S, start, midpoint)\n",
203 " else:\n",
204 " return binary_search(e, S, midpoint+1, end)\n",
205 " \n",
206 "def sort_list(L):\n",
207 " S = []\n",
208 " for e in L:\n",
209 " cp = binary_search(e, S, 0, len(S)) # Changed here\n",
210 " S.insert(cp, e)\n",
211 " return S\n",
212 " \n",
213 "N = 10000\n",
214 "L = [randint(0,10**9) for i in range(N)]\n",
215 "\n",
216 "t0 = time.process_time()\n",
217 "sort_list(L)\n",
218 "t1 = time.process_time()\n",
219 "\n",
220 "print(\"Running time:\", t1-t0)"
221 ]
222 },
223 {
224 "cell_type": "markdown",
225 "metadata": {},
226 "source": [
227 "# Fast exponentiation\n",
228 "\n",
229 "The following cell contains two functions for computing $a^n$ ($n$ non-negative integer): a slow one that runs in $O(n)$ and a fast one that runs in $O(\\log_2(n))$. We compare these two also with Python's built-in operator `**`.\n",
230 "\n",
231 "Complexity: $O(n)$ for the slow algorithm, $O(\\log_2(n))$ for the other two."
232 ]
233 },
234 {
235 "cell_type": "code",
236 "execution_count": 30,
237 "metadata": {},
238 "outputs": [
239 {
240 "name": "stdout",
241 "output_type": "stream",
242 "text": [
243 "2.71828179834636\n",
244 "2.7182817863957984\n",
245 "2.7182817983473577\n",
246 "Time for slow_power(): 3.234879998000004\n",
247 "Time for fast_power(): 9.059099999575437e-05\n",
248 "Time for Python's **: 0.00010159500000384014\n"
249 ]
250 }
251 ],
252 "source": [
253 "import time\n",
254 "\n",
255 "def slow_power(a, n):\n",
256 " r = 1\n",
257 " for i in range(n):\n",
258 " r = r * a\n",
259 " return r\n",
260 "\n",
261 "def fast_power(a, n):\n",
262 " if n == 0:\n",
263 " return 1\n",
264 " if n%2 == 0:\n",
265 " return fast_power(a*a, n//2)\n",
266 " else:\n",
267 " return a * fast_power(a, n-1)\n",
268 "\n",
269 "a = 1.00000001\n",
270 "n = 100000000\n",
271 "\n",
272 "t0 = time.process_time()\n",
273 "print(slow_power(a, n))\n",
274 "t1 = time.process_time()\n",
275 "print(fast_power(a, n))\n",
276 "t2 = time.process_time()\n",
277 "print(a**n)\n",
278 "t3 = time.process_time()\n",
279 "\n",
280 "print(\"Time for slow_power():\", t1-t0)\n",
281 "print(\"Time for fast_power():\", t2-t1)\n",
282 "print(\"Time for Python's **: \", t3-t2)"
283 ]
284 },
285 {
286 "cell_type": "markdown",
287 "metadata": {},
288 "source": [
289 "# Fast gcd\n",
290 "\n",
291 "Complexity: $O(\\log_2(n))$"
292 ]
293 },
294 {
295 "cell_type": "code",
296 "execution_count": 31,
297 "metadata": {},
298 "outputs": [
299 {
300 "name": "stdout",
301 "output_type": "stream",
302 "text": [
303 "126\n",
304 "Running time: 0.00017707599999994272\n"
305 ]
306 }
307 ],
308 "source": [
309 "import time\n",
310 "\n",
311 "def gcd(a, b):\n",
312 " if b == 0:\n",
313 " return a\n",
314 " else:\n",
315 " return gcd(b, a%b)\n",
316 "\n",
317 "t0 = time.process_time()\n",
318 "print(gcd(155275387236018, 572335397352432))\n",
319 "t1 = time.process_time()\n",
320 "\n",
321 "print(\"Running time:\", t1-t0)"
322 ]
323 },
324 {
325 "cell_type": "markdown",
326 "metadata": {},
327 "source": [
328 "# Fibonacci numbers\n",
329 "\n",
330 "In the following cell there are two functions that compute the $n$-th Fibonacci number. They are almost the same, but the second one memorizes the results in a list to avoid computing them multiple times, and it is much much faster.\n",
331 "\n",
332 "Complexity: $O\\left(\\left(\\frac{1+\\sqrt 5}{2}\\right)^n\\right)\\sim O(1.6^n)$ for the slow version, $O(n)$ for the fast version."
333 ]
334 },
335 {
336 "cell_type": "code",
337 "execution_count": null,
338 "metadata": {},
339 "outputs": [],
340 "source": [
341 "import time\n",
342 "\n",
343 "F_memorized = [-1] * (10**6)\n",
344 "\n",
345 "def F_slow(n):\n",
346 " if n <= 1:\n",
347 " return n\n",
348 " else:\n",
349 " return F_slow(n-1) + F_slow(n-2)\n",
350 " \n",
351 "def F_fast(n):\n",
352 " if F_memorized[n] == -1:\n",
353 " if n <= 1:\n",
354 " F_memorized[n] = n\n",
355 " else:\n",
356 " F_memorized[n] = F_fast(n-1) + F_fast(n-2)\n",
357 " \n",
358 " return F_memorized[n]\n",
359 "\n",
360 "n = 40\n",
361 "\n",
362 "t0 = time.process_time()\n",
363 "print(F_slow(n))\n",
364 "t1 = time.process_time()\n",
365 "print(F_fast(n))\n",
366 "t2 = time.process_time()\n",
367 "\n",
368 "print(\"Time for F_slow:\", t1-t0)\n",
369 "print(\"Time for F_fast:\", t2-t1)"
370 ]
371 }
372 ],
373 "metadata": {
374 "kernelspec": {
375 "display_name": "Python 3",
376 "language": "python",
377 "name": "python3"
378 },
379 "language_info": {
380 "codemirror_mode": {
381 "name": "ipython",
382 "version": 3
383 },
384 "file_extension": ".py",
385 "mimetype": "text/x-python",
386 "name": "python",
387 "nbconvert_exporter": "python",
388 "pygments_lexer": "ipython3",
389 "version": "3.8.5"
390 }
391 },
392 "nbformat": 4,
393 "nbformat_minor": 4
394}
diff --git a/src/Lecture7/slides/.ipynb_checkpoints/X1-ComputationalComplexity-notebook-checkpoint.ipynb b/src/Lecture7/slides/.ipynb_checkpoints/X1-ComputationalComplexity-notebook-checkpoint.ipynb
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1{
2 "cells": [
3 {
4 "cell_type": "markdown",
5 "metadata": {},
6 "source": [
7 "# Nested loops\n",
8 "\n",
9 "The following two functions compute sum and product of matrices, respectively.\n",
10 "\n",
11 "By counting the nested loops it is easy to see that `add()` is $O(n^2)$ while `prod()` is $O(n^3)$."
12 ]
13 },
14 {
15 "cell_type": "code",
16 "execution_count": 2,
17 "metadata": {},
18 "outputs": [
19 {
20 "name": "stdout",
21 "output_type": "stream",
22 "text": [
23 "Time for add: 0.005766554000000035\n",
24 "Time for prod: 1.3871021639999999\n"
25 ]
26 }
27 ],
28 "source": [
29 "from random import randint\n",
30 "import time\n",
31 "\n",
32 "def add(A, B):\n",
33 " S = [[0] * len(A) for i in range(len(A))]\n",
34 " for i in range(len(A)):\n",
35 " for j in range(len(A)):\n",
36 " S[i][j] = A[i][j] + B[i][j]\n",
37 " return S\n",
38 "\n",
39 "def prod(A, B):\n",
40 " S = [[0] * len(A) for i in range(len(A))]\n",
41 " for i in range(len(A)):\n",
42 " for j in range(len(A)):\n",
43 " for k in range(len(A)):\n",
44 " S[i][j] = S[i][j] + A[i][k] * B[k][j]\n",
45 " return S\n",
46 "\n",
47 "N = 200\n",
48 "A = [ [randint(0,100) for i in range(N)] for j in range(N) ]\n",
49 "B = [ [randint(0,100) for i in range(N)] for j in range(N) ]\n",
50 "\n",
51 "t0 = time.process_time()\n",
52 "add(A,B)\n",
53 "t1 = time.process_time()\n",
54 "prod(A,B)\n",
55 "t2 = time.process_time()\n",
56 "\n",
57 "print(\"Time for add: \", t1-t0)\n",
58 "print(\"Time for prod:\", t2-t1)"
59 ]
60 },
61 {
62 "cell_type": "markdown",
63 "metadata": {},
64 "source": [
65 "# Sorting a list, slow version\n",
66 "\n",
67 "The following code implements a slow version of the so-called *insertion sort* alogithm\n",
68 "\n",
69 "Complexity: $O(n^2)$."
70 ]
71 },
72 {
73 "cell_type": "code",
74 "execution_count": 1,
75 "metadata": {},
76 "outputs": [
77 {
78 "name": "stdout",
79 "output_type": "stream",
80 "text": [
81 "Running time: 1.1191449070000001\n"
82 ]
83 }
84 ],
85 "source": [
86 "from random import randint\n",
87 "import time\n",
88 "\n",
89 "def correct_position(e, S):\n",
90 " for i in range(len(S)):\n",
91 " if S[i] > e:\n",
92 " return i\n",
93 " return len(S)\n",
94 "\n",
95 "def sort_list(L):\n",
96 " S = []\n",
97 " for e in L:\n",
98 " cp = correct_position(e, S)\n",
99 " S.insert(cp, e)\n",
100 " return S\n",
101 "\n",
102 "N = 10000\n",
103 "L = [randint(0,10**9) for i in range(N)]\n",
104 "\n",
105 "t0 = time.process_time()\n",
106 "sort_list(L)\n",
107 "t1 = time.process_time()\n",
108 "\n",
109 "print(\"Running time:\", t1-t0)"
110 ]
111 },
112 {
113 "cell_type": "markdown",
114 "metadata": {},
115 "source": [
116 "# Binary search\n",
117 "\n",
118 "The following code implements a binary search.\n",
119 "\n",
120 "Complexity: $O(\\log_2(n))$"
121 ]
122 },
123 {
124 "cell_type": "code",
125 "execution_count": 3,
126 "metadata": {},
127 "outputs": [
128 {
129 "name": "stdout",
130 "output_type": "stream",
131 "text": [
132 "The correct position of e = 216197744 in L is:\n",
133 "... 216196218 216197540 e 216198673 216198962 ...\n",
134 "\n",
135 "Time for sorting: 0.26413054400000036\n",
136 "Time for searching: 9.616099999965044e-05\n"
137 ]
138 }
139 ],
140 "source": [
141 "from random import randint\n",
142 "import time\n",
143 "\n",
144 "def binary_search(e, S, start, end):\n",
145 " if start == end:\n",
146 " return start\n",
147 " midpoint = (start+end) // 2\n",
148 " if e < S[midpoint]:\n",
149 " return binary_search(e, S, start, midpoint)\n",
150 " else:\n",
151 " return binary_search(e, S, midpoint+1, end)\n",
152 " \n",
153 "N = 1000000\n",
154 "L = [randint(0,10**9) for i in range(N)]\n",
155 "e = randint(0,10**9)\n",
156 "\n",
157 "t0 = time.process_time()\n",
158 "L.sort() # Using Python's sort()\n",
159 "t1 = time.process_time()\n",
160 "i = binary_search(e, L, 0, len(L))\n",
161 "t2 = time.process_time()\n",
162 "print(\"The correct position of e =\", e, \"in L is:\")\n",
163 "print(\"...\", L[i-2], L[i-1], \"e\", L[i], L[i+1], \"...\")\n",
164 "print(\"\")\n",
165 "print(\"Time for sorting: \", t1-t0)\n",
166 "print(\"Time for searching:\", t2-t1)\n"
167 ]
168 },
169 {
170 "cell_type": "markdown",
171 "metadata": {},
172 "source": [
173 "# Sorting a list, fast version (with binary_search)\n",
174 "\n",
175 "The following code uses the function `binary_search()` above instead of `correct_position()` in our insertion sort algorithm.\n",
176 "\n",
177 "Complexity: $O(n\\log_2(n))$"
178 ]
179 },
180 {
181 "cell_type": "code",
182 "execution_count": 69,
183 "metadata": {},
184 "outputs": [
185 {
186 "name": "stdout",
187 "output_type": "stream",
188 "text": [
189 "Running time: 0.03710268399998995\n"
190 ]
191 }
192 ],
193 "source": [
194 "from random import randint\n",
195 "import time\n",
196 "\n",
197 "def binary_search(e, S, start, end):\n",
198 " if start == end:\n",
199 " return start\n",
200 " midpoint = (start+end) // 2\n",
201 " if e < S[midpoint]:\n",
202 " return binary_search(e, S, start, midpoint)\n",
203 " else:\n",
204 " return binary_search(e, S, midpoint+1, end)\n",
205 " \n",
206 "def sort_list(L):\n",
207 " S = []\n",
208 " for e in L:\n",
209 " cp = binary_search(e, S, 0, len(S)) # Changed here\n",
210 " S.insert(cp, e)\n",
211 " return S\n",
212 " \n",
213 "N = 10000\n",
214 "L = [randint(0,10**9) for i in range(N)]\n",
215 "\n",
216 "t0 = time.process_time()\n",
217 "sort_list(L)\n",
218 "t1 = time.process_time()\n",
219 "\n",
220 "print(\"Running time:\", t1-t0)"
221 ]
222 },
223 {
224 "cell_type": "markdown",
225 "metadata": {},
226 "source": [
227 "# Fast exponentiation\n",
228 "\n",
229 "The following cell contains two functions for computing $a^n$ ($n$ non-negative integer): a slow one that runs in $O(n)$ and a fast one that runs in $O(\\log_2(n))$. We compare these two also with Python's built-in operator `**`.\n",
230 "\n",
231 "Complexity: $O(n)$ for the slow algorithm, $O(\\log_2(n))$ for the other two."
232 ]
233 },
234 {
235 "cell_type": "code",
236 "execution_count": 30,
237 "metadata": {},
238 "outputs": [
239 {
240 "name": "stdout",
241 "output_type": "stream",
242 "text": [
243 "2.71828179834636\n",
244 "2.7182817863957984\n",
245 "2.7182817983473577\n",
246 "Time for slow_power(): 3.234879998000004\n",
247 "Time for fast_power(): 9.059099999575437e-05\n",
248 "Time for Python's **: 0.00010159500000384014\n"
249 ]
250 }
251 ],
252 "source": [
253 "import time\n",
254 "\n",
255 "def slow_power(a, n):\n",
256 " r = 1\n",
257 " for i in range(n):\n",
258 " r = r * a\n",
259 " return r\n",
260 "\n",
261 "def fast_power(a, n):\n",
262 " if n == 0:\n",
263 " return 1\n",
264 " if n%2 == 0:\n",
265 " return fast_power(a*a, n//2)\n",
266 " else:\n",
267 " return a * fast_power(a, n-1)\n",
268 "\n",
269 "a = 1.00000001\n",
270 "n = 100000000\n",
271 "\n",
272 "t0 = time.process_time()\n",
273 "print(slow_power(a, n))\n",
274 "t1 = time.process_time()\n",
275 "print(fast_power(a, n))\n",
276 "t2 = time.process_time()\n",
277 "print(a**n)\n",
278 "t3 = time.process_time()\n",
279 "\n",
280 "print(\"Time for slow_power():\", t1-t0)\n",
281 "print(\"Time for fast_power():\", t2-t1)\n",
282 "print(\"Time for Python's **: \", t3-t2)"
283 ]
284 },
285 {
286 "cell_type": "markdown",
287 "metadata": {},
288 "source": [
289 "# Fast gcd\n",
290 "\n",
291 "Complexity: $O(\\log_2(n))$"
292 ]
293 },
294 {
295 "cell_type": "code",
296 "execution_count": 31,
297 "metadata": {},
298 "outputs": [
299 {
300 "name": "stdout",
301 "output_type": "stream",
302 "text": [
303 "126\n",
304 "Running time: 0.00017707599999994272\n"
305 ]
306 }
307 ],
308 "source": [
309 "import time\n",
310 "\n",
311 "def gcd(a, b):\n",
312 " if b == 0:\n",
313 " return a\n",
314 " else:\n",
315 " return gcd(b, a%b)\n",
316 "\n",
317 "t0 = time.process_time()\n",
318 "print(gcd(155275387236018, 572335397352432))\n",
319 "t1 = time.process_time()\n",
320 "\n",
321 "print(\"Running time:\", t1-t0)"
322 ]
323 },
324 {
325 "cell_type": "markdown",
326 "metadata": {},
327 "source": [
328 "# Fibonacci numbers\n",
329 "\n",
330 "In the following cell there are two functions that compute the $n$-th Fibonacci number. They are almost the same, but the second one memorizes the results in a list to avoid computing them multiple times, and it is much much faster.\n",
331 "\n",
332 "Complexity: $O\\left(\\left(\\frac{1+\\sqrt 5}{2}\\right)^n\\right)\\sim O(1.6^n)$ for the slow version, $O(n)$ for the fast version."
333 ]
334 },
335 {
336 "cell_type": "code",
337 "execution_count": 37,
338 "metadata": {},
339 "outputs": [
340 {
341 "name": "stdout",
342 "output_type": "stream",
343 "text": [
344 "9227465\n",
345 "9227465\n",
346 "Time for F_slow: 2.3301570169999906\n",
347 "Time for F_fast: 8.848800000293977e-05\n"
348 ]
349 }
350 ],
351 "source": [
352 "import time\n",
353 "\n",
354 "F_memorized = [-1] * (10**6)\n",
355 "\n",
356 "def F_slow(n):\n",
357 " if n <= 1:\n",
358 " return n\n",
359 " else:\n",
360 " return F_slow(n-1) + F_slow(n-2)\n",
361 " \n",
362 "def F_fast(n):\n",
363 " if F_memorized[n] == -1:\n",
364 " if n <= 1:\n",
365 " F_memorized[n] = n\n",
366 " else:\n",
367 " F_memorized[n] = F_fast(n-1) + F_fast(n-2)\n",
368 " \n",
369 " return F_memorized[n]\n",
370 "\n",
371 "n = 35\n",
372 "\n",
373 "t0 = time.process_time()\n",
374 "print(F_slow(n))\n",
375 "t1 = time.process_time()\n",
376 "print(F_fast(n))\n",
377 "t2 = time.process_time()\n",
378 "\n",
379 "print(\"Time for F_slow:\", t1-t0)\n",
380 "print(\"Time for F_fast:\", t2-t1)"
381 ]
382 },
383 {
384 "cell_type": "code",
385 "execution_count": null,
386 "metadata": {},
387 "outputs": [],
388 "source": []
389 }
390 ],
391 "metadata": {
392 "kernelspec": {
393 "display_name": "Python 3",
394 "language": "python",
395 "name": "python3"
396 },
397 "language_info": {
398 "codemirror_mode": {
399 "name": "ipython",
400 "version": 3
401 },
402 "file_extension": ".py",
403 "mimetype": "text/x-python",
404 "name": "python",
405 "nbconvert_exporter": "python",
406 "pygments_lexer": "ipython3",
407 "version": "3.8.5"
408 }
409 },
410 "nbformat": 4,
411 "nbformat_minor": 4
412}
diff --git a/src/Lecture7/slides/.ipynb_checkpoints/X2-StudentsRequests-checkpoint.ipynb b/src/Lecture7/slides/.ipynb_checkpoints/X2-StudentsRequests-checkpoint.ipynb
new file mode 100644
index 0000000..a35bb8d
--- /dev/null
+++ b/src/Lecture7/slides/.ipynb_checkpoints/X2-StudentsRequests-checkpoint.ipynb
@@ -0,0 +1,165 @@
1{
2 "cells": [
3 {
4 "cell_type": "markdown",
5 "metadata": {},
6 "source": [
7 "# Diffie-Hellman key exchange\n",
8 "\n",
9 "The following is a simple implementation of the classic [Diffie-Hellman key exchange](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) cryptographic protocol."
10 ]
11 },
12 {
13 "cell_type": "code",
14 "execution_count": 9,
15 "metadata": {},
16 "outputs": [
17 {
18 "name": "stdout",
19 "output_type": "stream",
20 "text": [
21 "Public key: p = 20747 and g = 13428 \n",
22 "\n",
23 "[[ Alice's secret key: a = 12403 ]]\n",
24 "[[ Bob's secret key: b = 17642 ]] \n",
25 "\n",
26 "Alice sends h1 = 14710 to Bob\n",
27 "Bob sends h2 = 10680 to Alice \n",
28 "\n",
29 "Alice computed 10455 using h2 and her secret a\n",
30 "Bob computed 10455 using h1 and his secret b\n"
31 ]
32 }
33 ],
34 "source": [
35 "# Public information:\n",
36 "p = Primes()[10^3 + randint(1,10000)] # random prime\n",
37 "g = randint(2, p-1) # random integer\n",
38 "\n",
39 "print(\"Public key: p =\", p, \"and g =\", g, \"\\n\")\n",
40 "\n",
41 "a = randint(2, p-1) # Only Alice knows this\n",
42 "b = randint(2, p-1) # Only Bob knows this\n",
43 "\n",
44 "print(\"[[ Alice's secret key: a =\", a, \"]]\")\n",
45 "print(\"[[ Bob's secret key: b =\", b, \"]]\", \"\\n\")\n",
46 "\n",
47 "h1 = (g^a) % p # Alice sends this to Bob\n",
48 "h2 = (g^b) % p # Bob sends this to Alice\n",
49 "\n",
50 "print(\"Alice sends h1 =\", h1, \"to Bob\")\n",
51 "print(\"Bob sends h2 =\", h2, \"to Alice\", \"\\n\")\n",
52 "\n",
53 "secret_a = (h2^a) % p # Alice can compute this because she knows a\n",
54 "secret_b = (h1^b) % p # Bob can compute this because he knows b\n",
55 "\n",
56 "print(\"Alice computed\", secret_a, \"using h2 and her secret a\")\n",
57 "print(\"Bob computed\", secret_b, \"using h1 and his secret b\")"
58 ]
59 },
60 {
61 "cell_type": "markdown",
62 "metadata": {},
63 "source": [
64 "## General Diffie-Hellman\n",
65 "\n",
66 "The following code is an implementation of a generic Diffie-Hellman key exchange protocol that uses a group $G$ instead of $(\\mathbb Z/p \\mathbb Z)^\\times$."
67 ]
68 },
69 {
70 "cell_type": "code",
71 "execution_count": 24,
72 "metadata": {},
73 "outputs": [
74 {
75 "name": "stdout",
76 "output_type": "stream",
77 "text": [
78 "Public key:\n",
79 "G = Additive abelian group isomorphic to Z/171 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x + 156 over Finite Field of size 157 \n",
80 "g = (155 : 60 : 1) \n",
81 "\n",
82 "[[ Alice's secret key: a = 141 ]]\n",
83 "[[ Bob's secret key: b = 158 ]] \n",
84 "\n",
85 "Alice sends h1 = (29 : 125 : 1) to Bob\n",
86 "Bob sends h2 = (60 : 59 : 1) to Alice \n",
87 "\n",
88 "Alice computed (109 : 94 : 1) using h2 and her secret a\n",
89 "Bob computed (109 : 94 : 1) using h1 and his secret b\n"
90 ]
91 }
92 ],
93 "source": [
94 "def genericDH(G):\n",
95 " if G.cardinality() == 1:\n",
96 " print(\"Group is trivial, can't do anything\")\n",
97 " return\n",
98 " g = G.random_element()\n",
99 " while g == G.identity(): # Make sure g is not trivial\n",
100 " g = G.random_element()\n",
101 " \n",
102 " print(\"Public key:\\nG =\", G, \"\\ng =\", g, \"\\n\")\n",
103 " \n",
104 " a = randint(2, G.exponent()-1) # Only Alice knows this\n",
105 " b = randint(2, G.exponent()-1) # Only Bob knows this\n",
106 "\n",
107 " print(\"[[ Alice's secret key: a =\", a, \"]]\")\n",
108 " print(\"[[ Bob's secret key: b =\", b, \"]]\", \"\\n\")\n",
109 " \n",
110 " # \"Ternary operator\", I did not explain this\n",
111 " # https://docs.python.org/3/reference/expressions.html#conditional-expressions\n",
112 " h1 = g^a if G.is_multiplicative() else a*g # Alice sends this to Bob\n",
113 " h2 = g^b if G.is_multiplicative() else b*g # Bob sends this to Alice\n",
114 "\n",
115 " print(\"Alice sends h1 =\", h1, \"to Bob\")\n",
116 " print(\"Bob sends h2 =\", h2, \"to Alice\", \"\\n\")\n",
117 " \n",
118 " secret_a = h2^a if G.is_multiplicative() else a*h2 # Alice can compute this because she knows a\n",
119 " secret_b = h1^b if G.is_multiplicative() else b*h1 # Bob can compute this because he knows b\n",
120 "\n",
121 " print(\"Alice computed\", secret_a, \"using h2 and her secret a\")\n",
122 " print(\"Bob computed\", secret_b, \"using h1 and his secret b\")\n",
123 " \n",
124 "E = EllipticCurve(GF(157), [1,-1])\n",
125 "G = E.abelian_group()\n",
126 "genericDH(G)"
127 ]
128 },
129 {
130 "cell_type": "markdown",
131 "metadata": {},
132 "source": [
133 "# Numerical methods for PDEs"
134 ]
135 },
136 {
137 "cell_type": "code",
138 "execution_count": null,
139 "metadata": {},
140 "outputs": [],
141 "source": []
142 }
143 ],
144 "metadata": {
145 "kernelspec": {
146 "display_name": "SageMath 9.0",
147 "language": "sage",
148 "name": "sagemath"
149 },
150 "language_info": {
151 "codemirror_mode": {
152 "name": "ipython",
153 "version": 3
154 },
155 "file_extension": ".py",
156 "mimetype": "text/x-python",
157 "name": "python",
158 "nbconvert_exporter": "python",
159 "pygments_lexer": "ipython3",
160 "version": "3.8.5"
161 }
162 },
163 "nbformat": 4,
164 "nbformat_minor": 4
165}
diff --git a/src/Lecture7/slides/.ipynb_checkpoints/X2-StudentsRequests-notebook-checkpoint.ipynb b/src/Lecture7/slides/.ipynb_checkpoints/X2-StudentsRequests-notebook-checkpoint.ipynb
new file mode 100644
index 0000000..ccd6a21
--- /dev/null
+++ b/src/Lecture7/slides/.ipynb_checkpoints/X2-StudentsRequests-notebook-checkpoint.ipynb
@@ -0,0 +1,298 @@
1{
2 "cells": [
3 {
4 "cell_type": "markdown",
5 "metadata": {},
6 "source": [
7 "# Diffie-Hellman key exchange\n",
8 "\n",
9 "The following is a simple implementation of the classic [Diffie-Hellman key exchange](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) cryptographic protocol."
10 ]
11 },
12 {
13 "cell_type": "code",
14 "execution_count": 1,
15 "metadata": {},
16 "outputs": [
17 {
18 "name": "stdout",
19 "output_type": "stream",
20 "text": [
21 "Public key: p = 75521 and g = 58258 \n",
22 "\n",
23 "[[ Alice's secret key: a = 22794 ]]\n",
24 "[[ Bob's secret key: b = 69773 ]] \n",
25 "\n",
26 "Alice sends h1 = 31067 to Bob\n",
27 "Bob sends h2 = 54398 to Alice \n",
28 "\n",
29 "Alice computed 30031 using h2 and her secret a\n",
30 "Bob computed 30031 using h1 and his secret b\n"
31 ]
32 }
33 ],
34 "source": [
35 "# Public information:\n",
36 "p = Primes()[10^3 + randint(1,10000)] # random prime\n",
37 "g = randint(2, p-1) # random integer\n",
38 "\n",
39 "print(\"Public key: p =\", p, \"and g =\", g, \"\\n\")\n",
40 "\n",
41 "a = randint(2, p-1) # Only Alice knows this\n",
42 "b = randint(2, p-1) # Only Bob knows this\n",
43 "\n",
44 "print(\"[[ Alice's secret key: a =\", a, \"]]\")\n",
45 "print(\"[[ Bob's secret key: b =\", b, \"]]\", \"\\n\")\n",
46 "\n",
47 "h1 = (g^a) % p # Alice sends this to Bob\n",
48 "h2 = (g^b) % p # Bob sends this to Alice\n",
49 "\n",
50 "print(\"Alice sends h1 =\", h1, \"to Bob\")\n",
51 "print(\"Bob sends h2 =\", h2, \"to Alice\", \"\\n\")\n",
52 "\n",
53 "secret_a = (h2^a) % p # Alice can compute this because she knows a\n",
54 "secret_b = (h1^b) % p # Bob can compute this because he knows b\n",
55 "\n",
56 "print(\"Alice computed\", secret_a, \"using h2 and her secret a\")\n",
57 "print(\"Bob computed\", secret_b, \"using h1 and his secret b\")"
58 ]
59 },
60 {
61 "cell_type": "markdown",
62 "metadata": {},
63 "source": [
64 "## General Diffie-Hellman\n",
65 "\n",
66 "The following code is an implementation of a generic Diffie-Hellman key exchange protocol that uses a group $G$ instead of $(\\mathbb Z/p \\mathbb Z)^\\times$."
67 ]
68 },
69 {
70 "cell_type": "code",
71 "execution_count": 2,
72 "metadata": {},
73 "outputs": [
74 {
75 "name": "stdout",
76 "output_type": "stream",
77 "text": [
78 "Public key:\n",
79 "G = Additive abelian group isomorphic to Z/171 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x + 156 over Finite Field of size 157 \n",
80 "g = (53 : 90 : 1) \n",
81 "\n",
82 "[[ Alice's secret key: a = 145 ]]\n",
83 "[[ Bob's secret key: b = 65 ]] \n",
84 "\n",
85 "Alice sends h1 = (150 : 80 : 1) to Bob\n",
86 "Bob sends h2 = (4 : 58 : 1) to Alice \n",
87 "\n",
88 "Alice computed (28 : 28 : 1) using h2 and her secret a\n",
89 "Bob computed (28 : 28 : 1) using h1 and his secret b\n"
90 ]
91 }
92 ],
93 "source": [
94 "def genericDH(G):\n",
95 " if G.cardinality() == 1:\n",
96 " print(\"Group is trivial, can't do anything\")\n",
97 " return\n",
98 " g = G.random_element()\n",
99 " while g == G.identity(): # Make sure g is not trivial\n",
100 " g = G.random_element()\n",
101 " \n",
102 " print(\"Public key:\\nG =\", G, \"\\ng =\", g, \"\\n\")\n",
103 " \n",
104 " a = randint(2, G.exponent()-1) # Only Alice knows this\n",
105 " b = randint(2, G.exponent()-1) # Only Bob knows this\n",
106 "\n",
107 " print(\"[[ Alice's secret key: a =\", a, \"]]\")\n",
108 " print(\"[[ Bob's secret key: b =\", b, \"]]\", \"\\n\")\n",
109 " \n",
110 " # \"Ternary operator\", I did not explain this\n",
111 " # https://docs.python.org/3/reference/expressions.html#conditional-expressions\n",
112 " h1 = g^a if G.is_multiplicative() else a*g # Alice sends this to Bob\n",
113 " h2 = g^b if G.is_multiplicative() else b*g # Bob sends this to Alice\n",
114 "\n",
115 " print(\"Alice sends h1 =\", h1, \"to Bob\")\n",
116 " print(\"Bob sends h2 =\", h2, \"to Alice\", \"\\n\")\n",
117 " \n",
118 " secret_a = h2^a if G.is_multiplicative() else a*h2 # Alice can compute this because she knows a\n",
119 " secret_b = h1^b if G.is_multiplicative() else b*h1 # Bob can compute this because he knows b\n",
120 "\n",
121 " print(\"Alice computed\", secret_a, \"using h2 and her secret a\")\n",
122 " print(\"Bob computed\", secret_b, \"using h1 and his secret b\")\n",
123 " \n",
124 "E = EllipticCurve(GF(157), [1,-1])\n",
125 "G = E.abelian_group()\n",
126 "genericDH(G)"
127 ]
128 },
129 {
130 "cell_type": "markdown",
131 "metadata": {},
132 "source": [
133 "# Numerical methods for differential equations\n",
134 "\n",
135 "## Euler's method (ODE)\n",
136 "\n",
137 "In sage you can use [`ode_solver()`](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/ode.html) to solve any ordinary differential equation by hand, but Euler's method is very simple to implement by hand:"
138 ]
139 },
140 {
141 "cell_type": "code",
142 "execution_count": 4,
143 "metadata": {},
144 "outputs": [
145 {
146 "data": {
147 "image/png": 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148 "text/plain": [
149 "Graphics object consisting of 2 graphics primitives"
150 ]
151 },
152 "execution_count": 4,
153 "metadata": {},
154 "output_type": "execute_result"
155 }
156 ],
157 "source": [
158 "var('y')\n",
159 "\n",
160 "def euler_desolve(f, x0, y0, x1):\n",
161 " n = 5\n",
162 " h = (x1-x0)/n\n",
163 " S = []\n",
164 " Y = [y0]\n",
165 " for i in range(n+1):\n",
166 " S.append(x0 + i*h)\n",
167 " Y.append(N( Y[i] + h*f(S[i], Y[i]) ))\n",
168 " return S, Y\n",
169 "\n",
170 "f(x,y) = y\n",
171 "x0 = -1\n",
172 "x1 = 2\n",
173 "y0 = e^(-1)\n",
174 "\n",
175 "S, Y = euler_desolve(f, x0, y0, x1)\n",
176 "plot(e^x, -1, 2) + line([(S[i], Y[i]) for i in range(len(S))], color='red', marker='o', markersize=2)"
177 ]
178 },
179 {
180 "cell_type": "markdown",
181 "metadata": {},
182 "source": [
183 "Sage also has an `eulers_method()` function \"for pedagogical purposes only\":"
184 ]
185 },
186 {
187 "cell_type": "code",
188 "execution_count": 5,
189 "metadata": {},
190 "outputs": [
191 {
192 "name": "stdout",
193 "output_type": "stream",
194 "text": [
195 " x y h*f(x,y)\n",
196 " -1 0.367879441171442 0.0367879441171442\n",
197 "-0.900000000000000 0.404667385288587 0.0404667385288587\n",
198 "-0.800000000000000 0.445134123817445 0.0445134123817445\n",
199 "-0.700000000000000 0.489647536199190 0.0489647536199190\n",
200 "-0.600000000000000 0.538612289819109 0.0538612289819109\n",
201 "-0.500000000000000 0.592473518801020 0.0592473518801020\n",
202 "-0.400000000000000 0.651720870681122 0.0651720870681122\n",
203 "-0.300000000000000 0.716892957749234 0.0716892957749234\n",
204 "-0.200000000000000 0.788582253524157 0.0788582253524157\n",
205 "-0.100000000000000 0.867440478876573 0.0867440478876573\n",
206 "-1.38777878078145e-16 0.954184526764230 0.0954184526764230\n",
207 "0.0999999999999999 1.04960297944065 0.104960297944065\n",
208 "0.200000000000000 1.15456327738472 0.115456327738472\n",
209 "0.300000000000000 1.27001960512319 0.127001960512319\n",
210 "0.400000000000000 1.39702156563551 0.139702156563551\n",
211 "0.500000000000000 1.53672372219906 0.153672372219906\n",
212 "0.600000000000000 1.69039609441897 0.169039609441897\n",
213 "0.700000000000000 1.85943570386086 0.185943570386086\n",
214 "0.800000000000000 2.04537927424695 0.204537927424695\n",
215 "0.900000000000000 2.24991720167165 0.224991720167165\n",
216 "1.00000000000000 2.47490892183881 0.247490892183881\n",
217 "1.10000000000000 2.72239981402269 0.272239981402269\n",
218 "1.20000000000000 2.99463979542496 0.299463979542496\n",
219 "1.30000000000000 3.29410377496746 0.329410377496746\n",
220 "1.40000000000000 3.62351415246420 0.362351415246420\n",
221 "1.50000000000000 3.98586556771062 0.398586556771062\n",
222 "1.60000000000000 4.38445212448168 0.438445212448168\n",
223 "1.70000000000000 4.82289733692985 0.482289733692985\n",
224 "1.80000000000000 5.30518707062284 0.530518707062284\n",
225 "1.90000000000000 5.83570577768512 0.583570577768512\n",
226 "2.00000000000000 6.41927635545363 0.641927635545363\n"
227 ]
228 }
229 ],
230 "source": [
231 "# Usage: eulers_method(f, x0, y0, h, x1)\n",
232 "eulers_method(f, -1, N(e^(-1)), 0.1, 2)"
233 ]
234 },
235 {
236 "cell_type": "markdown",
237 "metadata": {},
238 "source": [
239 "## Solving the heat equation with a finite difference method"
240 ]
241 },
242 {
243 "cell_type": "code",
244 "execution_count": null,
245 "metadata": {},
246 "outputs": [],
247 "source": [
248 "def heat_fdm(u0j, u1j, ui0):\n",
249 " m, n = len(u0j)-1, len(ui0)-1\n",
250 " k, h = 1/m, 1/n\n",
251 " \n",
252 " u = [[0] * (m+1) for i in range(n+1)]\n",
253 " for j in range(m+1):\n",
254 " u[0][j] = u0j[j]\n",
255 " for j in range(m+1):\n",
256 " u[n][j] = u1j[j]\n",
257 " for i in range(n+1):\n",
258 " u[i][0] = ui0[i]\n",
259 " \n",
260 " for j in range(0,m):\n",
261 " for i in range(1,n):\n",
262 " u[i][j+1] = (k/(h*h)) * (u[i+1][j] - 2*u[i][j] + u[i-1][j]) + u[i][j]\n",
263 " \n",
264 " return u\n",
265 "\n",
266 "n, m = 20, 20\n",
267 "u0j = [10 - (j/m)*10 for j in range(m+1)] # One extreme goes from hot to cold\n",
268 "u1j = [(j/m)*10 for j in range(m+1)] # The other does the opposite\n",
269 "ui0 = [10 - (i/m)*10 for i in range(0,n+1)]\n",
270 "\n",
271 "u = heat_fdm(u0j, u1j, ui0)\n",
272 "for t in range(m+1):\n",
273 " show(line([(i/n, u[i][t]) for i in range(n+1)], ymin=-1, ymax =12))"
274 ]
275 }
276 ],
277 "metadata": {
278 "kernelspec": {
279 "display_name": "SageMath 9.0",
280 "language": "sage",
281 "name": "sagemath"
282 },
283 "language_info": {
284 "codemirror_mode": {
285 "name": "ipython",
286 "version": 3
287 },
288 "file_extension": ".py",
289 "mimetype": "text/x-python",
290 "name": "python",
291 "nbconvert_exporter": "python",
292 "pygments_lexer": "ipython3",
293 "version": "3.8.5"
294 }
295 },
296 "nbformat": 4,
297 "nbformat_minor": 4
298}
diff --git a/src/Lecture7/slides/X1-ComputationalComplexity.aux b/src/Lecture7/slides/X1-ComputationalComplexity.aux
new file mode 100644
index 0000000..b0463b4
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1\documentclass[11pt]{beamer}
2\usetheme{Madrid}
3\usepackage[utf8]{inputenc}
4\usepackage{amsmath}
5
6\usepackage{color}
7\usepackage{listings}
8\usepackage{mathtools}
9\usepackage{tikz-cd}
10\usepackage{adjustbox}
11
12\definecolor{myblue}{rgb}{0,0,0.5}
13\lstset{
14 language=Python,
15 tabsize=4,
16 basicstyle=\footnotesize,
17 keywordstyle=\bf\color{myblue},
18 commentstyle=\it\color{gray},
19 numbers=left,
20 numbersep=3pt,
21 numberstyle=\tiny\color{gray},
22}
23
24\author[\texttt{sebastiano.tronto@uni.lu}]{Sebastiano Tronto}
25\title[Computational Complexity]%
26{Why is my code slow?}
27\logo{\includegraphics[scale=0.1]{img/unilu.jpg}}
28%\institute{University of Luxembourg}
29
30\date{2021-05-21}
31
32\begin{document}
33
34\begin{frame}
35 \titlepage
36\end{frame}
37
38\begin{frame}{Computational Complexity}
39 \begin{itemize}
40 \item \textbf{Goal:}
41 estimate the running {\color{blue}time} of a program
42 \item \textbf{How:}
43 count the {\color{blue}basic steps} that an
44 {\color{blue}algorithm} takes to complete
45 \item \textbf{Why}:
46 find the \emph{bottleneck} of your program, make it faster
47 \end{itemize}
48
49 \vspace{0.5cm}
50 Our analysis should not depend on the hardware
51\end{frame}
52
53\begin{frame}{Algorithm}
54 \begin{definition}
55 \emph{An algorithm is a sequence of {\color{blue}steps} needed to
56 solve a {\color{blue}class of problems}. }
57 \end{definition}
58
59 \begin{definition}[alternative]
60 \emph{An algorithm is a sequence of steps that takes
61 an input satisfying certain conditions and produces an output
62 satisfying other conditions.}
63 \end{definition}
64\end{frame}
65
66\begin{frame}{Sorting a list}
67 \begin{block}{Class of problems}
68 Sort a list $L$ of numbers in increasing order.
69 \end{block}
70
71 \begin{block}{Algorithm}
72 \begin{enumerate}
73 \item Let $S$ be an empty list.
74 \item Take an element from $L$ an insert it in $S$ in its correct
75 position.
76 \item Repeat step $2$ until $L$ is empty.
77 \item Return $S$.
78 \end{enumerate}
79 \end{block}
80\end{frame}
81
82\begin{frame}{Sorting a list}
83\begin{itemize}
84 \item It solves a \emph{class} of problems: works for any list
85 \item The specific steps to sort the list $[3,7,1]$ are not an algorithm
86 \item Input conditions: must be a list of numbers
87 \item Output conditions: same numbers in increasing order
88\end{itemize}
89\end{frame}
90
91\begin{frame}{How to write an algorithm}
92 \begin{itemize}
93 \item \textbf{Human language}:
94 \begin{itemize}
95 \item Easy to understand
96 \item Not precise
97 \end{itemize}
98
99 \vspace{0.3cm}
100 \item \textbf{Computer code}:
101 \begin{itemize}
102 \item Can be executed by computers
103 \item Precise
104 \item From very low level (machine code) to high level
105 (Python, \dots)
106 \end{itemize}
107 \end{itemize}
108
109 %\vspace{0.5cm}
110 %To what \emph{level of detail}?
111\end{frame}
112
113\begin{frame}{Basic steps}
114 \begin{itemize}
115 %\item Strictly speaking, only CPU instructions are \emph{basic}
116 %\item In practice:%, we consider basic:
117 % \begin{itemize}
118 \item Arithmetic operations $+,-,*,//,\%$
119 \item Relational operations $==, !=, >, <,\dots$
120 \item Memory access (read/write variable)
121 % \end{itemize}
122 \end{itemize}
123
124 \vspace{0.5cm}
125 \textbf{Warning:}
126 Depends on data type (integer, floating point, string,\dots)
127 %\begin{itemize}
128 % \item Depends on data type (integer, floating point, string,\dots)
129 % \item There are non-basic instructions such as \texttt{sort()}
130 %\end{itemize}
131\end{frame}
132
133\begin{frame}{Running time}
134 \begin{itemize}
135 \item Depends on computer power, programming language, compiler\dots
136 %\item Not all basic steps are equal
137 \item ``Big O'' notation: an algorithm runs in time $O(f(n))$ if, when
138 run with input of size $n$, it takes about $c\cdot f(n)$ steps
139 \item Algorithm A is \emph{asymptotically faster} than algorithm B if
140 it is faster \textbf{for $n$ large enough}
141 \item Rule of thumb: $10^7\sim10^9$ basic steps per second
142 \end{itemize}
143\end{frame}
144
145\begin{frame}{Asymptotical analysis vs constant factors}
146 \includegraphics[scale=0.7]{img/plot1.png}
147\end{frame}
148
149\begin{frame}{Asymptotical analysis vs constant factors}
150 \includegraphics[scale=0.7]{img/plot2.png}
151\end{frame}
152
153\begin{frame}{Asymptotical analysis vs constant factors}
154 \includegraphics[scale=0.7]{img/plot3.png}
155\end{frame}
156
157\begin{frame}{Asymptotical analysis vs constant factors}
158 \includegraphics[scale=0.7]{img/plot4.png}
159\end{frame}
160
161\begin{frame}{Asymptotical analysis vs constant factors}
162 \includegraphics[scale=0.7]{img/plot5.png}
163\end{frame}
164
165%\begin{frame}{title}
166%graphs here, uncomment
167%\end{frame}
168
169\begin{frame}{Basic complexity analysis}
170
171 Easy things to do:
172
173 \vspace{0.3cm}
174 \begin{itemize}
175 \item Check documentation for ``non-basic steps''
176 \begin{itemize}
177 \item Example: check Sage's \href{https://doc.sagemath.org/html/en/reference/rings\_standard/sage/rings/integer.html\#sage.rings.integer.Integer.is\_prime}{\texttt{is\_prime()}} (redirects to PARI \href{https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic\_functions.html\#se:isprime}{\texttt{isprime()}})
178 \end{itemize}
179
180 \vspace{0.3cm}
181 \item Count nested loops
182 \begin{itemize}
183 \item How many times is a step repeated?
184 \end{itemize}
185 \end{itemize}
186\end{frame}
187
188{\setbeamertemplate{logo}{}
189\begin{frame}[fragile]{Nested loops - matrix sum and product}
190\begin{lstlisting}
191def add(A, B):
192 n = len(A)
193 S = [[0] * n for i in range(n)]
194 for i in range(0, n):
195 for j in range(0, n):
196 S[i][j] = A[i][j] + B[i][j]
197 return S
198\end{lstlisting}
199
200\vspace{0.5cm}
201\begin{lstlisting}
202def prod(A, B):
203 n = len(A)
204 S = [[0] * n for i in range(n)]
205 for i in range(0, n):
206 for j in range(0, n):
207 for k in range(0, n):
208 S[i][j] = S[i][j] + A[i][k]*B[k][j]
209 return S
210\end{lstlisting}
211\end{frame}
212}
213
214\begin{frame}{Nested loops - matrix sum and product}
215 \begin{itemize}
216 \item \texttt{add} is $O(n^2)$ (two loops)
217 \item \texttt{prod} is $O(n^3)$ (three loops)
218 \end{itemize}
219
220 \vspace{0.3cm}
221 \textbf{Fun fact:} there are faster algorithms for matrix multiplication,
222 for example \href{https://en.wikipedia.org/wiki/Strassen_algorithm}%
223 {Strassen's algorithm}.
224\end{frame}
225
226\begin{frame}[fragile]{Sorting a list}
227\begin{lstlisting}
228def correct_position(e, S):
229 for i in range(0, len(S)):
230 if S[i] > e:
231 return i
232 return len(S)
233
234def sort_list(L):
235 S = []
236 for e in L:
237 cp = correct_position(e, S)
238 S.insert(cp, e)
239 return S
240\end{lstlisting}
241\end{frame}
242
243\begin{frame}{Sorting a list}
244 \begin{itemize}
245 \item Complexity of \texttt{correct\_position()}:
246 \begin{itemize}
247 %\item best case $O(1)$
248 \item worst case $O($\texttt{len(S)}$)$
249 \item average $O($\texttt{len(S)}$)$
250 \end{itemize}
251
252 \vspace{0.3cm}
253 \item Complexity of \texttt{sort\_list} (here $n=$\texttt{len(L)}):
254 \begin{align*}
255 %\sum_{i=0}^{n-1} O(1) = O(n) && \text{best case}\\
256 \sum_{i=0}^{n-1} O(i) = O(n^2)% && \text{average/worst}
257 \end{align*}
258 (it calls \texttt{correct\_position()} $n$ times).
259 \end{itemize}
260\end{frame}
261
262\begin{frame}{Sorting a list}
263 \begin{itemize}
264 \item For which lists does the ``best case'' happen?
265 \item For which lists does the ``worst case'' happen?
266 \item How large can $n$ be for \texttt{sort\_list()} to run
267 in under a second?
268 \end{itemize}
269\end{frame}
270
271\begin{frame}{Sorting a list}
272 How to improve our code?
273 \begin{itemize}
274 \item Improve \texttt{correct\_position()}
275 \item Take advantage of the fact that $S$ is always sorted
276 \end{itemize}
277\end{frame}
278
279\begin{frame}{Binary search}
280 \begin{block}{Algorithm}
281 \textbf{Input:} a \emph{sorted} list $S$ and a value $e$.
282 \begin{enumerate}
283 \item If the list is empty, you have found the position of $e$
284 \item Otherwise, compare $e$ to the middle element $m$ of $S$
285 \begin{itemize}
286 \item If $e<m$, repeat from (1) on the first half of $S$
287 \item Otherwise, repeat from (1) on the second half of $S$
288 \end{itemize}
289 \end{enumerate}
290 \end{block}
291\end{frame}
292
293\begin{frame}[fragile]{Binary search}
294\begin{lstlisting}
295# Return position of e in L
296def binary_search(e, S, start, end):
297 if start == end:
298 return start
299 midpoint = (end+start)//2
300 if e < S[midpoint]:
301 return binary_search(e, S, start, midpoint)
302 else:
303 return binary_search(e, S, midpoint+1, end)
304\end{lstlisting}
305\end{frame}
306
307\begin{frame}{Binary search - example 1}
308 Searching for \texttt{e}$=2$:
309 \begin{align*}
310 \only<1>{
311 \underbrace{
312 \overset{{\color{blue}
313 \substack{\mathclap{\texttt{start}=0}\\\downarrow}}}{-2}
314 \quad 0\quad 1\quad 3\quad
315 \overset{\substack{\mathclap{\texttt{midpoint}=4}\\\downarrow}}{5}
316 \quad 6\quad 7\quad 9\quad 12
317 }\quad
318 \overset{{\color{red}
319 \substack{\mathclap{\texttt{end}=9}\\\downarrow}}}{\phantom{0}}
320 }
321 \only<2>{
322 \underbrace{
323 \overset{{\color{blue}
324 \substack{\mathclap{\texttt{start}=0}\\\downarrow}}}{-2}
325 \quad 0\quad
326 \overset{\substack{\mathclap{\texttt{midpoint}=2}\\\\\downarrow}}%
327 {1}
328 \quad 3
329 }\quad
330 \overset{{\color{red}
331 \substack{\mathclap{\texttt{end}=4}\\\downarrow}}}{5}
332 \quad 6\quad 7\quad 9\quad 12\quad \phantom{0}
333 }
334 \only<3>{
335 -2\quad 0\quad 1\quad
336 \underbrace{
337 \overset{
338 \substack{
339 \mathclap{
340 {\color{blue}\texttt{start}}=\texttt{midpoint}=3}\\\\
341 {\color{blue}\downarrow}
342 }
343 }{3}
344 } \quad
345 \overset{{\color{red}
346 \substack{\mathclap{\texttt{end}=4}\\\downarrow}}}{5}
347 \quad 6\quad 7\quad 9\quad 12\quad \phantom{0}
348 }
349 \only<4>{
350 -2\quad 0\quad 1\quad
351 \overset{
352 \substack{
353 \mathclap{
354 {\color{blue}\texttt{start}}=
355 {\color{red}\texttt{end}}=3}\\\downarrow}}{3}
356 \quad 5 \quad 6\quad 7\quad 9\quad 12\quad \phantom{0}
357 }
358 \end{align*}
359 \only<1>{{\color{blue}$e<5$}$\implies$ check left half}
360 \only<2>{{\color{red}$e>1$}$\implies$ check right half}
361 \only<3>{{\color{blue}$e<3$}$\implies$ check left half}
362 \only<4>{\texttt{start}=\texttt{end}, done}
363\end{frame}
364
365\begin{frame}{Binary search - example 2}
366 Searching for \texttt{e}$=11$:
367 \begin{align*}
368 \only<1>{
369 \underbrace{
370 \overset{{\color{blue}
371 \substack{\mathclap{\texttt{start}=0}\\\downarrow}}}{-2}
372 \quad 0\quad 1\quad 3\quad
373 \overset{\substack{\mathclap{\texttt{midpoint}=4}\\\downarrow}}{5}
374 \quad 6\quad 7\quad 9\quad 12
375 }\quad
376 \overset{{\color{red}
377 \substack{\mathclap{\texttt{end}=9}\\\downarrow}}}{\phantom{0}}
378 }
379 \only<2>{
380 -2 \quad 0\quad 1 \quad 3 \quad 5 \quad
381 \underbrace{
382 \overset{{\color{blue}
383 \substack{\mathclap{\texttt{start}=5}\\\downarrow}}}{6}
384 \quad 7 \quad
385 \overset{\substack{\mathclap{\texttt{midpoint}=7}\\\\\downarrow}}%
386 {9}
387 \quad 12
388 }\quad
389 \overset{{\color{red}
390 \substack{\mathclap{\texttt{end}=9}\\\downarrow}}}{\phantom{0}}
391 }
392 \only<3>{
393 -2\quad 0\quad 1\quad 3\quad 5\quad 6\quad 7\quad 9\quad
394 \underbrace{
395 \overset{
396 \substack{
397 \mathclap{
398 {\color{blue}\texttt{start}}=\texttt{midpoint}=8}\\\\
399 {\color{blue}\downarrow}
400 }
401 }{12}
402 } \quad
403 \overset{{\color{red}
404 \substack{\mathclap{\texttt{end}=9}\\\downarrow}}}{\phantom{0}}
405 }
406 \only<4>{
407 -2\quad 0\quad 1\quad 3\quad 5\quad 6\quad 7\quad 9\quad
408 \overset{
409 \substack{
410 \mathclap{
411 {\color{blue}\texttt{start}}=
412 {\color{red}\texttt{end}}=8}\\\downarrow}}{12}
413 }
414 \end{align*}
415 \only<1>{{\color{red}$e>5$}$\implies$ check right half}
416 \only<2>{{\color{red}$e>9$}$\implies$ check right half}
417 \only<3>{{\color{blue}$e<11$}$\implies$ check left half}
418 \only<4>{\texttt{start}=\texttt{end}, done}
419\end{frame}
420
421\begin{frame}{Binary search}
422 \begin{itemize}
423 \item Works only if the list is sorted
424 \item Complexity $O(\log_2(n))$: at every step we cut the list in half
425 \item Recursive, \emph{divide et impera}
426 \end{itemize}
427\end{frame}
428
429\begin{frame}[fragile]{Sorting a list - binary search version}
430\begin{lstlisting}
431def sort_list(L):
432 S = []
433 for e in L:
434 cp = binary_search(e, S, 0, len(S)) # This changed
435 S.insert(cp, e)
436 return S
437\end{lstlisting}
438 \vspace{0.3cm}
439 \begin{itemize}
440 \item Complexity: \[\sum_{i=0}^{n-1} O(\log_2(i)) = O(n\log_2(n))\]\\
441 (it calls \texttt{binary\_search} $n$ times).
442 \end{itemize}
443\end{frame}
444
445\begin{frame}{Fast exponentiation}
446 \begin{block}{Algorithm / formula}
447 \begin{align*}
448 a^n=
449 \begin{cases}
450 1 & \text{if }n=0,\\
451 (a\cdot a)^{\frac n2} & \text{if $n$ is even},\\
452 a\cdot a^{n-1} & \text{if $n$ is odd.}
453 \end{cases}
454 \end{align*}
455 \end{block}
456\end{frame}
457
458\begin{frame}[fragile]{Fast exponentiation}
459\begin{lstlisting}
460# Compute a^n (n>=0 integer)
461def power(a, n):
462 if n == 0:
463 return 1
464 if n % 2 == 0: # n is even
465 return power(a*a, n//2)
466 else: # n is odd
467 return a*power(a, n-1)
468\end{lstlisting}
469\end{frame}
470
471\begin{frame}{Fast exponentiation}
472
473 \begin{itemize}
474 \item Complexity: $O(\log_2(n))$ (after $2$ steps, $n$ is halved)
475 \item Python's operator $**$ does something similar
476 \item Naive algorithm (one loop): $O(n)$
477 \end{itemize}
478\end{frame}
479
480
481\begin{frame}[fragile]{Fast $\gcd$}
482 \begin{block}{Algorithm / formula}
483 \begin{align*}
484 \gcd(a,b) =
485 \begin{cases}
486 a & \text{if }b=0,\\
487 \gcd(b,a\bmod b) & \text{otherwise.}
488 \end{cases}
489 \end{align*}
490 \end{block}
491
492\begin{columns}
493\column{0.5\textwidth}
494\begin{lstlisting}
495def gcd(a, b):
496 if b == 0:
497 return a
498 else:
499 return gcd(b, a%b)
500\end{lstlisting}
501
502\column{0.5\textwidth}
503\begin{itemize}
504 \item After $2$ steps, $a$ is halved $\implies$ complexity $O(\log_2(a))$
505\end{itemize}
506\end{columns}
507\end{frame}
508
509\begin{frame}{Recursion}
510 \begin{itemize}
511 \item These examples use \emph{recursion}
512 (a function that calls itself)
513 \item If it calls itself more than once, it is slow
514 (\emph{exponential} complexity!)
515 \end{itemize}
516\end{frame}
517
518\begin{frame}[fragile]{Fibonacci numbers}
519
520 \begin{block}{Algorithm / formula}
521 \begin{align*}
522 F(n) =
523 \begin{cases}
524 n & \text{if }n\leq1,\\
525 F(n-1)+F(n-2) & \text{otherwise.}
526 \end{cases}
527 \end{align*}
528 \end{block}
529
530\vspace{0.5cm}
531\begin{lstlisting}
532def F(n):
533 if n <= 1:
534 return n
535 else:
536 return F(n-1) + F(n-2)
537\end{lstlisting}
538\end{frame}
539
540\begin{frame}[fragile]{Fibonacci}
541 \begin{adjustbox}{scale={0.85}{0.9},center}
542 \begin{tikzcd}[column sep=1mm]
543 & & & & & & & & F(5) \ar[drrr] \ar[dlll]\\
544 & & & & & F(4)\ar[dll]\ar[dr] & & & & & & F(3) \ar[dl] \ar[dr]\\
545 & & & F(3) \ar[dl]\ar[dr] & & & F(2) \ar[dr]\ar[dl]
546 & & & & F(2) \ar[dl]\ar[dr] & & F(1) \\
547 & & F(2) \ar[dl]\ar[dr] & & F(1) & F(1) & & F(0) & & F(1) & & F(0)\\
548 & F(1) & & F(0)
549 \end{tikzcd}
550 \end{adjustbox}
551\end{frame}
552
553\begin{frame}{Fibonacci}
554 \begin{itemize}
555 \item Complexity: almost $O(2^n)$ (actually $O(\varphi^n)$
556 with $\varphi=\frac{1+\sqrt 5}{2}\sim 1.6$)
557 \item But some values are computed many times!
558 \item Optimization: memorize previously computed values
559 \end{itemize}
560\end{frame}
561
562\begin{frame}[fragile]{Fibonacci with memorization}
563\begin{lstlisting}
564# List with memorized values, N is the largest possible
565N = 10**6
566F_memorized = [-1] * N
567
568def F(n):
569 if F_memorized[n] == -1:
570 if n <= 1:
571 F_memorized[n] = n
572 else:
573 F_memorized[n] = F(n-1) + F(n-2)
574
575 return F_memorized[n]
576\end{lstlisting}
577\end{frame}
578
579\begin{frame}[fragile]{Fibonacci with memorization}
580 \begin{adjustbox}{scale={0.85}{0.9},center}
581 \begin{tikzcd}[column sep=1mm]
582 & & & & & & & & F(5) \ar[drrr] \ar[dlll]\\
583 & & & & & F(4)\ar[dll]\ar[dr] & & & & & & {\color{blue}F(3)}\\
584 & & & F(3) \ar[dl]\ar[dr] & & & {\color{blue}F(2)}\\
585 & & F(2) \ar[dl]\ar[dr] & & {\color{blue}F(1)} \\
586 & F(1) & & F(0)
587 \end{tikzcd}
588 \end{adjustbox}
589\end{frame}
590
591\begin{frame}{Fibonacci with memorization}
592 \begin{itemize}
593 \item Complexity: $O(n)$, huge improvement!
594 \item Further improvement (but still $O(n)$): dynamic programming
595 \item Pay attention to memory usage
596 \end{itemize}
597\end{frame}
598
599\begin{frame}{References}
600 \begin{itemize}
601 \item Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and
602 Clifford Stein -
603 \href{https://en.wikipedia.org/wiki/Introduction\_to\_Algorithms}%
604 {\emph{Introductions to Algorithms}}
605 \end{itemize}
606\end{frame}
607
608\end{document}
diff --git a/src/Lecture7/slides/X1-ComputationalComplexity.toc b/src/Lecture7/slides/X1-ComputationalComplexity.toc
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1\frametitle{Fibonacci with memorization}
2\begin{adjustbox}{scale={0.85}{0.9},center}
3 \begin{tikzcd}[column sep=1mm]
4 & & & & & & & & F(5) \ar[drrr] \ar[dlll]\\
5 & & & & & F(4)\ar[dll]\ar[dr] & & & & & & {\color{blue}F(3)}\\
6 & & & F(3) \ar[dl]\ar[dr] & & & {\color{blue}F(2)}\\
7 & & F(2) \ar[dl]\ar[dr] & & {\color{blue}F(1)} \\
8 & F(1) & & F(0)
9 \end{tikzcd}
10 \end{adjustbox}
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1\documentclass[11pt]{beamer}
2\usetheme{Madrid}
3\usepackage[utf8]{inputenc}
4\usepackage{amsmath}
5
6\usepackage{svg}
7\usepackage{color}
8\usepackage{listings}
9\usepackage{mathtools}
10\usepackage{tikz-cd}
11\usepackage{adjustbox}
12
13\definecolor{myblue}{rgb}{0,0,0.5}
14\lstset{
15 language=Python,
16 tabsize=4,
17 basicstyle=\footnotesize,
18 keywordstyle=\bf\color{myblue},
19 commentstyle=\it\color{gray},
20 numbers=left,
21 numbersep=3pt,
22 numberstyle=\tiny\color{gray},
23}
24
25\author[\texttt{sebastiano.tronto@uni.lu}]{Sebastiano Tronto}
26\title[Students requests]%
27{Students requests}
28\logo{\includegraphics[scale=0.1]{img/unilu.jpg}}
29%\institute{University of Luxembourg}
30
31\date{2021-05-21}
32
33\begin{document}
34
35\begin{frame}
36 \titlepage
37\end{frame}
38
39\begin{frame}[plain]
40 \begin{center} {\Huge More cryptography} \end{center}
41\end{frame}
42
43\begin{frame}{Cryptography}
44 What we have seen:
45
46 \vspace{0.3cm}
47 \begin{itemize}
48 \item \textbf{RSA:}
49 sending messages using a private key / public key pair
50 \item \textbf{Flip-a-coin:}
51 cryptographic ``proof'' that the opponent is not cheating
52 \end{itemize}
53\end{frame}
54
55\begin{frame}{Cryptography}
56 \begin{itemize}
57 \item Rely on integer factorization being hard
58
59 \vspace{0.3cm}
60 \textbf{Example:} the best-known factorization algorithm
61 (\href{https://en.wikipedia.org/wiki/General\_number\_field\_sieve}%
62 {\emph{General number field sieve}}) has complexity
63 \begin{align*}
64 \sim O\left(
65 e^{\sqrt[3]{\frac{64}{9}\log_2n\cdot(\log_2\log_2n)^2}}
66 \right)
67 \end{align*}
68
69 Factoring a number with $300$ digits:
70 \begin{itemize}
71 \item Your laptop: $10^{13}$ billion years
72 \item Best supercomputer: $13$ billion years
73 (age of the universe)
74 \end{itemize}
75 \end{itemize}
76\end{frame}
77
78\begin{frame}{Symmetric and asymmetric cryptography}
79 \begin{itemize}
80 \item Our examples are \emph{asymmetric}: different public/private keys
81 \item Safe against eavesdroppers
82 \item Symmetric protocols can be faster and simpler, but you need
83 a secure way to exchange a key
84 \end{itemize}
85\end{frame}
86
87\begin{frame}{Diffie-Hellman key exchange}
88 \begin{itemize}
89 \item Generate a ``password'' without communicating it directly
90 \item It can then be used for symmetric cryptography
91 \item Based on a different hard problem:
92 \href{https://en.wikipedia.org/wiki/Discrete\_logarithm}%
93 {\emph{discrete logarithm}}
94 \end{itemize}
95\end{frame}
96
97\begin{frame}{Diffie-Hellman key exchange}
98 \begin{itemize}
99 \item Alice and Bob agree on a prime number $p$ and an integer $g$
100 \item Alice picks an integer $a$ and sends $(g^a\bmod p)$ to Bob
101 \item Bob picks an integer $b$ and sends $(g^b\bmod p)$ to Alice
102 \item Alice can compute $(g^b)^a\bmod p$ and Bob can compute
103 $(g^a)^b\bmod p$. This is their shared secret (key).
104 \end{itemize}
105\end{frame}
106
107\begin{frame}{Diffie-Hellman with colors (from Wikipedia)}
108 \begin{center}\includesvg[scale=0.45]{img/DH}\end{center}
109\end{frame}
110
111\begin{frame}{Diffie-Hellman key exchange}
112 \begin{itemize}
113 \item Knowing $h$ and $a$, it is hard to find $g$ such that
114 $g^a \bmod p =h$ (discrete logarithm problem)
115 \item Very simple, many variants
116 \item Any group can be used, e.g. Elliptic Curves (see
117\href{https://en.wikipedia.org/wiki/Elliptic-curve_Diffie\%E2\%80\%93Hellman}%
118 {Wikipedia: elliptic-curve Diffie-Hellman})
119 \end{itemize}
120\end{frame}
121
122
123\begin{frame}[plain]
124 \begin{center} {\Huge Numerical methods for PDEs} \end{center}
125\end{frame}
126
127\begin{frame}{Solving partial differential equations}
128 \begin{itemize}
129 \item Very, very hard
130 \item Very important in practical applications (physics and such)
131 \item Approximations are necessary, might as well use numerical methods
132 \end{itemize}
133\end{frame}
134
135\begin{frame}{Numerical methods for ODEs}
136 \begin{block}{Problem}
137 Given $f(x,y)$, $x_0$ and $y_0$, find an approximation
138 for $y(x)$ such that
139 \begin{align*}
140 \begin{cases}
141 y'(x) = f(x, y(x))\\
142 y(x_0) =y_0
143 \end{cases}
144 \end{align*}
145 \end{block}
146
147 \begin{block}{Approximation}
148 We can describe $y(x)$ in an interval $[x_0,x_1]$ by giving the
149 (approximate) values $y(s_0)$, \dots, $y(s_n)$ for many
150 values of $s_i\in [x_0, x_1]$.
151 \end{block}
152\end{frame}
153
154\begin{frame}{Euler's method}
155 \begin{block}{Idea}
156 For $h$ small
157 \begin{align*}
158 y'(x)\approx\frac{y(x+h)-y(x)}{h}
159 \end{align*}
160 which implies
161 \begin{align*}
162 y(x+h) \approx y(x) + h\cdot f(x, y(x))
163 \end{align*}
164 \end{block}
165\end{frame}
166
167\begin{frame}{Euler's method}
168 \begin{block}{Algorithm}
169 \textbf{Input:} the data $f(x,y)$, $x_0$, $y_0$ and $x_1$ describing
170 the problem and the desired range for the solution.
171
172 \vspace{0.3cm}
173 \textbf{Output:} $x_0=s_0 < s_1 < \dots < s_n=x_1$ and
174 $y_0, \dots, y_n$ such that $y_i\approx y(s_i)$.
175
176 \vspace{0.3cm}
177 \begin{enumerate}
178 \item Choose a value $n$ and let
179 $h=\frac{x_1-x_0}{n}$ and $s_i=x_0+ih$
180 \item For $i=0,\dots, n-1$ compute
181 $y_{i+1}=y_i+h\cdot f(s_i, y_i)$
182 \item Return $s_0, \dots, s_n$ and $y_0, \dots, y_n$
183 \end{enumerate}
184 \end{block}
185\end{frame}
186
187\begin{frame}{Euler's method}
188 \begin{itemize}
189 \item Very simple and fast
190 \item Generalization for higher-order equations: Runge-Kutta methods
191 \item A similar idea works for some PDEs
192 \end{itemize}
193\end{frame}
194
195\begin{frame}{The heat equation (PDE)}
196 \begin{align*}
197 \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2} +
198 \frac{\partial^2 u}{\partial x_2^2} + \cdots +
199 \frac{\partial^2 u}{\partial x_n^2}
200 \end{align*}
201
202 Where
203 \[u(x_1,x_2,\dots,x_n,t): \mathbb R^n\times \mathbb R_+\to \mathbb R\]
204 describes the quantity of heat at the point $(x_1,\dots x_n)$ at time $t$.
205
206 \vspace{0.3cm} It appears also outside thermodynamics: mathematical finance
207 (\href{https://en.wikipedia.org/wiki/Black\%E2\%80\%93Scholes\_equation}%
208 {Black-Scholes equation}), quantum mechanics
209 (\href{https://en.wikipedia.org/wiki/Schr\%C3\%B6dinger\_equation}%
210 {Schrödinger equation}), image analysis\dots
211\end{frame}
212
213\begin{frame}{A simple case ($n=1$, in $[0,1]^2$)}
214 \begin{block}{Problem}
215 Given $u_0(t)$, $u_1(t)$ and $u^0(x)$, find an approximation
216 for $u(x,t)$ such that
217 \begin{align*}
218 \begin{cases}
219 \frac{\partial u}{\partial t} =
220 \frac{\partial^2 u}{\partial x^2} \\
221 u(0,t) = u_{(0)}(t) \quad \text{(boundary condition)}\\
222 u(1,t) = u_{(1)}(t) \quad \text{(boundary condition)}\\
223 u(x,0) = u^0(x) \quad \text{(initial condition)}
224 \end{cases}
225 \end{align*}
226 \end{block}
227
228 \begin{block}{Approximation}
229 Values $u_i^j\approx u(s_i, r^j)$ for
230 $(s_i,r^j)\in [0,1]\times [0,1]$
231 \end{block}
232\end{frame}
233
234\begin{frame}{Idea}
235 For $k$ small:
236 \begin{align*}
237 \frac{\partial u(x,t)}{\partial t} \approx \frac{u(x,t+k)-u(x,t)}{k}\\
238 \end{align*}
239 For $h$ small (left limit + right limit):
240 \begin{align*}
241 \frac{\partial^2 u(x,t)}{\partial x^2} &\approx
242 \frac{\partial}{\partial x}\left(
243 \frac{u(x,t) - u(x-h,t)}{h}
244 \right)\\
245 &\approx \frac1h\left(
246 \frac{\partial u(x,t)}{\partial x} -
247 \frac{\partial u(x-h,t)}{\partial x}
248 \right)\\
249 &\approx \frac1h\left(
250 \frac{u(x+h,t) - u(x,t)}{h} - \frac{u(x,t)-u(x-h,t)}{h}
251 \right)\\
252 &\approx \frac{u(x+h,t)-2u(x,t)+u(x-h,t)}{h^2}
253 \end{align*}
254\end{frame}
255
256\begin{frame}{Idea}
257 From the equation
258 \begin{align*}
259 \frac{u_i^{j+1}-u_i^j}{k}= \frac{u_{i+1}^j-2u_{i}^j+u_{i-1}^j}{h^2}
260 \end{align*}
261 we find the formula
262 \begin{align*}
263 u_i^{j+1} = \frac{k}{h^2}\left(u_{i+1}^j - 2u_i^j + u_{i-1}^j\right)
264 + u_i^j
265 \end{align*}
266\end{frame}
267
268\begin{frame}{Finite difference method for the heat equation}
269 \begin{block}{Algorithm}
270 \textbf{Input:} $u_{(0)}^j$, $u_{(1)}^j$ (boundary)
271 and $u_i^0$ (initial).
272
273 \vspace{0.3cm}
274 \textbf{Output:} values $u_i^j$ approximating a solution.
275
276 \vspace{0.3cm}
277 \begin{enumerate}
278 \item Let $m=\operatorname{len}(u_0)-1$,
279 $n=\operatorname{len}(u^0)-1$ and $k=1/m$, $h=1/n$
280 %\begin{align*}
281 % \begin{array}{cccc}
282 % k=\frac{t_1-t_0}{m}, & h=\frac{x_1-x_0}{n}, &
283 % r^j = t_0 +jk, & s_i = x_0+ih
284 % \end{array}
285 %\end{align*}
286 \item For $j=0,\dots, m-1$ do the following:
287 \begin{itemize}
288 \item For $i=1,\dots, n-1$ compute
289 \begin{align*}
290 u_i^{j+1} = \frac{k}{h^2}\left(u_{i+1}^j -
291 2u_i^j + u_{i-1}^j\right) + u_i^j
292 \end{align*}
293 \end{itemize}
294 \item Return the $u_i^j$
295 \end{enumerate}
296 \end{block}
297\end{frame}
298
299\begin{frame}{Other PDEs}
300 \begin{itemize}
301 \item In general, there is no generic method
302 \item You might need to write specific code for your equation
303 \item Some packages exists
304 (e.g. \href{https://wiki.octave.org/Fem-fenics}{fem-fenics} for
305 \href{https://www.gnu.org/software/octave/index}{Gnu Octave})
306 \end{itemize}
307\end{frame}
308
309\end{document}
diff --git a/src/Lecture7/slides/X2-StudentsRequests.toc b/src/Lecture7/slides/X2-StudentsRequests.toc
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/src/Lecture7/slides/X2-StudentsRequests.toc
diff --git a/src/Lecture7/slides/X2-StudentsRequests.vrb b/src/Lecture7/slides/X2-StudentsRequests.vrb
new file mode 100644
index 0000000..57dfae6
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@@ -0,0 +1,10 @@
1\frametitle{Fibonacci with memorization}
2\begin{adjustbox}{scale={0.85}{0.9},center}
3 \begin{tikzcd}[column sep=1mm]
4 & & & & & & & & F(5) \ar[drrr] \ar[dlll]\\
5 & & & & & F(4)\ar[dll]\ar[dr] & & & & & & {\color{blue}F(3)}\\
6 & & & F(3) \ar[dl]\ar[dr] & & & {\color{blue}F(2)}\\
7 & & F(2) \ar[dl]\ar[dr] & & {\color{blue}F(1)} \\
8 & F(1) & & F(0)
9 \end{tikzcd}
10 \end{adjustbox}
diff --git a/src/Lecture7/slides/img/DH.svg b/src/Lecture7/slides/img/DH.svg
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+++ b/src/Lecture7/slides/img/DH.svg
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181</svg> \ No newline at end of file
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@@ -0,0 +1,58 @@
1%% Creator: Inkscape inkscape 0.92.5, www.inkscape.org
2%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
3%% Accompanies image file 'DH_svg-tex.pdf' (pdf, eps, ps)
4%%
5%% To include the image in your LaTeX document, write
6%% \input{<filename>.pdf_tex}
7%% instead of
8%% \includegraphics{<filename>.pdf}
9%% To scale the image, write
10%% \def\svgwidth{<desired width>}
11%% \input{<filename>.pdf_tex}
12%% instead of
13%% \includegraphics[width=<desired width>]{<filename>.pdf}
14%%
15%% Images with a different path to the parent latex file can
16%% be accessed with the `import' package (which may need to be
17%% installed) using
18%% \usepackage{import}
19%% in the preamble, and then including the image with
20%% \import{<path to file>}{<filename>.pdf_tex}
21%% Alternatively, one can specify
22%% \graphicspath{{<path to file>/}}
23%%
24%% For more information, please see info/svg-inkscape on CTAN:
25%% http://tug.ctan.org/tex-archive/info/svg-inkscape
26%%
27\begingroup%
28 \makeatletter%
29 \providecommand\color[2][]{%
30 \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
31 \renewcommand\color[2][]{}%
32 }%
33 \providecommand\transparent[1]{%
34 \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
35 \renewcommand\transparent[1]{}%
36 }%
37 \providecommand\rotatebox[2]{#2}%
38 \newcommand*\fsize{\dimexpr\f@size pt\relax}%
39 \newcommand*\lineheight[1]{\fontsize{\fsize}{#1\fsize}\selectfont}%
40 \ifx\svgwidth\undefined%
41 \setlength{\unitlength}{319.9125bp}%
42 \ifx\svgscale\undefined%
43 \relax%
44 \else%
45 \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
46 \fi%
47 \else%
48 \setlength{\unitlength}{\svgwidth}%
49 \fi%
50 \global\let\svgwidth\undefined%
51 \global\let\svgscale\undefined%
52 \makeatother%
53 \begin{picture}(1,1.50094365)%
54 \lineheight{1}%
55 \setlength\tabcolsep{0pt}%
56 \put(0,0){\includegraphics[width=\unitlength,page=1]{DH_svg-tex.pdf}}%
57 \end{picture}%
58\endgroup%

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