-
Notifications
You must be signed in to change notification settings - Fork 32
/
euler-0014.cpp
150 lines (137 loc) · 5.29 KB
/
euler-0014.cpp
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
// ////////////////////////////////////////////////////////
// # Title
// Longest Collatz sequence
//
// # URL
// https://projecteuler.net/problem=14
// http://euler.stephan-brumme.com/14/
//
// # Problem
// The following iterative sequence is defined for the set of positive integers:
//
// if `n` is even: `n \to n/2`
// if `n` is odd: `n \to 3n + 1`
//
// Using the rule above and starting with 13, we generate the following sequence:
// `13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1`
//
// It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms.
// Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.
//
// Which starting number, under one million, produces the longest chain?
//
// __NOTE:__ Once the chain starts the terms are allowed to go above one million.
//
// # Solved by
// Stephan Brumme
// February 2017
//
// # Algorithm
// The longest Collatz chain below five million contains 597 elements (and starts with 3732423).
// A brute-force algorithm solves this problem within a half a second.
//
// A smarter approach is to cache all chain lengths we encounter along the way.
// My function ''steps(x)'' tries to look up the result for ''x'' in a ''cache'':
// - if it succeeds, just return that cached value
// - if it fails, we do one step in the Collatz sequence and call ''step'' recursively
//
// The code runs approx. 10x faster now (but needs about 10 MByte RAM more compared to the brute-force version).
//
// # Note
// I observed a max. recursion depth of 363 (incl. memoization). This shouldn't be a problem for most computers.
//
// # Alternative
// There are surprisingly few "thresholds", where the Collatz chain length is longer than anything we have seen before:
// > 1, 2, 3, 6, 7, 9, 18, 25, 27, 54,
// > 73, 97, 129, 171, 231, 313, 327, 649, 703, 871,
// > 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529,
// > 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631,
// > 410011, 511935, 626331, __837799__,1117065,1501353,1723519,2298025,3064033,3542887,
// > 3732423
//
// This sequence is all we need to solve the original problem (well, this list contains all numbers up to 5000000 but we only need up to 1000000).
// You can easily see that 837799 is the largest number less than one million.
//
// # Hackerrank
// I compute the Collatz sequences "on-demand":
// if the input value ''x'' exceeds ''maxTested'' then all still unknown Collatz sequences up to ''x'' will be analyzed.
// Whenever a sequence is at least as long as the longest sequence we observed so far, then I have to update ''longest''.
//
// This data structure ''longest'' is supposed to hold the values shown above.
// However, the modified Hackerrank problem asks for the __largest__ number with a given Collatz chain length
// but those numbers refers to the __smallest__.
//
// Therefore I have to insert all numbers where
// ''length >= longest.rbegin()->second'' instead of
// ''length > longest.rbegin()->second''
//
// In the end, my ''std::map'' contains a few more values:
// > 1, 2, 3, 6, 7, 9, 18, 19, 25, 27,
// > 54, 55, 73, 97, 129, 171, 231, 235, 313, 327,
// > 649, 654, 655, 667, 703, 871, 1161, 2223, 2322, 2323,
// > 2463, 2919, 3711, 6171, 10971, 13255, 17647, 17673, 23529, 26623,
// > 34239, 35497, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631,
// > 410011, 511935, 626331, 837799,1117065,1126015,1501353,1564063,1723519,2298025,
// > 3064033,3542887,3732423
#include <iostream>
#include <vector>
#include <map>
// memoize all paths length for n up to 5000000
const size_t MaxN = 5000000 + 2;
// we could change MaxN at will:
// it just affects performance, not the result
// identify chain lengths we haven't computed so far
const int Unknown = -1;
// store chain lengths
std::vector<short> cache(MaxN, Unknown);
// recursively count steps of Collatz sequence
unsigned short steps(unsigned long long x)
{
// finished recursion ?
if (x == 1)
return 1;
// try to use cached result
if (x < cache.size() && cache[x] != Unknown)
return cache[x];
// next step
long long next;
if (x % 2 == 0)
next = x / 2;
else
next = 3 * x + 1;
// deeper recursion
auto result = 1 + steps(next);
if (x < cache.size())
cache[x] = result;
return result;
}
int main()
{
// [smallest number] => [chain length]
std::map<unsigned int, unsigned int> longest;
// highest number analyzed so far
unsigned int maxTested = 1;
longest[maxTested] = 1; // obvious case
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int x;
std::cin >> x;
// compute remaining chain lengths
for (; maxTested <= x; maxTested++)
{
// get chain length
auto length = steps(maxTested);
// at least as long as anything we have seen before ?
if (length >= longest.rbegin()->second)
longest[maxTested] = length;
}
// find next longest chain for numbers bigger than x
auto best = longest.upper_bound(x);
// and go one step back
best--;
std::cout << best->first << std::endl;
}
return 0;
}