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Add Floyd's algorithm illustration
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‎source_code/README.md‎

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@@ -43,4 +43,7 @@ certain subtle points.
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errors.
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1. `iterative_binary_trees.ipynb`: using iteration rather than recursion to traverse
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binary trees.
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1. `quad_tree.ipynb`: implementation of a quad tree and comparison to naive approach.
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1. `quad_tree.ipynb`: implementation of a quad tree and comparison to naive
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approach.
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1. `cycles.ipynb`: find cycles in periodic functions. Naive vs. approach using
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dictionaries, vs. Floyd's algorithm.

‎source_code/cycles.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "72050593-cd1e-4b99-bf85-7a748b813694",
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"metadata": {},
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"outputs": [],
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"source": [
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"def find_cycles_naive(func, start_value, max_iters):\n",
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" history = [start_value]\n",
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" for generation in range(1, 1 + max_iters):\n",
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" value = func(history[-1])\n",
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" if value in history:\n",
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" offset = history.index(value)\n",
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" return offset, generation - offset + 1\n",
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" else:\n",
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" history.append(value)\n",
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" return max_iters, None"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 43,
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"id": "1973e2d5-f59d-439c-9bd3-b6381ac13299",
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"metadata": {},
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"outputs": [],
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"source": [
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"def collatz_conjecture(n):\n",
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" return n//2 if n % 2 == 0 else 3*n + 1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 49,
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"id": "8cf319fb-cc1a-49b7-a63e-97972ce768c7",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(60, 3)"
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]
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},
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"execution_count": 49,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"find_cycles_naive(collatz_conjecture, 1023, 100)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "92990251-8f03-46cb-9206-e71274ec87ba",
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"metadata": {},
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"outputs": [],
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"source": [
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"def find_cycles_dict(func, start_value, max_iters):\n",
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" history = {start_value: 0}\n",
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" value = start_value\n",
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" for generation in range(1, 1 + max_iters):\n",
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" value = func(value)\n",
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" if value in history:\n",
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" offset = history[value]\n",
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" return offset, generation - offset\n",
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" else:\n",
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" history[value] = generation\n",
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" return max_iters, None"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 61,
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"id": "a7e3de97-3031-4aff-b799-fec419ba6173",
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(60, 3)"
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]
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},
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"execution_count": 61,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"find_cycles_dict(collatz_conjecture, 1023, 100)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"id": "aa224337-7847-41da-8859-d285c994c528",
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"metadata": {},
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"outputs": [],
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"source": [
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"def find_cycles_floyd(func, start_value, max_iters):\n",
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" tortoise = func(start_value)\n",
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" hare = func(tortoise)\n",
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" while tortoise != hare:\n",
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" tortoise = func(tortoise)\n",
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" hare = func(func(hare))\n",
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" offset = 0\n",
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" tortoise = start_value\n",
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" while tortoise != hare:\n",
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" tortoise = func(tortoise)\n",
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" hare = func(hare)\n",
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" offset += 1\n",
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" length = 1\n",
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" hare = func(tortoise)\n",
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" while tortoise != hare:\n",
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" hare = func(hare)\n",
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" length += 1\n",
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" return offset, length"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 67,
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"id": "3639aefe-2e2b-4342-aebb-7736c64c8645",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(60, 3)"
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]
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},
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"execution_count": 67,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"find_cycles_floyd(collatz_conjecture, 1023, 100)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 71,
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"id": "42eed8d2-3443-4636-840e-6731350a686b",
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"metadata": {},
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"outputs": [],
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"source": [
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"for n in range(1024):\n",
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" naive_cycle = find_cycles_naive(collatz_conjecture, n, 100_000)\n",
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" dict_cycle = find_cycles_dict(collatz_conjecture, n, 100_000)\n",
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" floyd_cycle = find_cycles_floyd(collatz_conjecture, n, 100_000)\n",
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" assert naive_cycle == dict_cycle\n",
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" assert dict_cycle == floyd_cycle"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 69,
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"id": "7e4d214c-ba72-4007-98ba-dd7b61ed18e1",
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"metadata": {},
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"outputs": [],
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"source": [
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"def show_series(func, start_value, max_iters):\n",
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" value = start_value\n",
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" for _ in range(max_iters):\n",
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" print(value)\n",
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" value = func(value)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 76,
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"id": "ee8d5f5c-1257-420e-9e5c-e78900da3c1b",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"17.2 µs ± 248 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n"
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]
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}
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],
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"source": [
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"%timeit find_cycles_naive(collatz_conjecture, 1023, 100_000)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 77,
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"id": "eec36b86-2c86-4eda-8700-066ce1d011de",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"6.46 µs ± 136 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n"
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]
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}
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],
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"source": [
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"%timeit find_cycles_dict(collatz_conjecture, 1023, 100_000)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 78,
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"id": "511e6192-4dde-4bd8-9ade-d7838b5eccff",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"15.7 µs ± 289 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n"
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]
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}
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],
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"source": [
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"%timeit find_cycles_floyd(collatz_conjecture, 1023, 100_000)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 80,
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"id": "ada23862-9d69-4ccb-9309-5ddfc1d07645",
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"metadata": {},
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"outputs": [],
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"source": [
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"offsets = [find_cycles_dict(collatz_conjecture, n, 100_000)[0] for n in range(2**20 - 1)]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 50,
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"id": "2a28f9ba-a284-441d-8c0f-8e7c7365af6f",
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"metadata": {},
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"outputs": [],
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"source": [
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"def generate_periodic_function(offset, length):\n",
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" def internal(n):\n",
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" if n < offset - 1:\n",
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" return n + 1\n",
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" return offset + ((n + 1 - offset) % length)\n",
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" return internal"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 51,
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"id": "b34cb35a-9949-4187-94be-420196511c8c",
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"metadata": {},
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"outputs": [],
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"source": [
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"f = generate_periodic_function(5, 3)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 52,
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"id": "2a7e97c2-92bb-4ef7-9771-b21f9f791fa5",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1\n",
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"2\n",
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"3\n",
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"4\n",
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"5\n",
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"6\n",
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"7\n",
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"5\n",
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"6\n",
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"7\n",
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"5\n",
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"6\n",
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"7\n",
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"5\n",
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"6\n",
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"7\n",
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"5\n",
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"6\n",
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"7\n",
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"5\n"
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]
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}
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],
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"source": [
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"for n in range(20):\n",
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" print(f(n))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 53,
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"id": "fd92a91b-d1e7-4bf1-a7c9-7fb3b3d60f20",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(5, 3)"
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]
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},
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"execution_count": 53,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"find_cycles_dict(f, 0, 100)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 54,
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"id": "8be3dec2-736c-4617-b7cd-e016770e28fd",
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"metadata": {},
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"outputs": [],
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"source": [
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"for offset in range(10):\n",
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" for length in range(1, 5):\n",
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" f = generate_periodic_function(offset, length)\n",
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" assert (offset, length) == find_cycles_floyd(f, 0, 0)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "a7e10960-0a9c-447d-8ba2-edfe1900bc19",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

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