在数字数组中，每个数字出现偶数次，只有一个数字出现奇数次。我们需要找到那个数字(这个问题之前讨论过 on Stack Overflow )。
这是一个用 3 种不同方法解决问题的解决方案——两种方法是 O(N)(hash_set 和 hash_map)，一种是 O(NlogN)(排序)。然而，对任意大的输入进行分析表明排序更快，并且随着输入的增加而变得越来越快(相比之下)。
实现或复杂性分析有什么问题，为什么 O(NlogN) 方法更快？
#include <algorithm>
#include <chrono>
#include <cmath>
#include <iostream>
#include <functional>
#include <string>
#include <vector>
#include <unordered_set>
#include <unordered_map>
using std::cout;
using std::chrono::high_resolution_clock;
using std::chrono::milliseconds;
using std::endl;
using std::string;
using std::vector;
using std::unordered_map;
using std::unordered_set;
class ScopedTimer {
public:
ScopedTimer(const string& name)
: name_(name), start_time_(high_resolution_clock::now()) {}
~ScopedTimer() {
cout << name_ << " took "
<< std::chrono::duration_cast<milliseconds>(
high_resolution_clock::now() - start_time_).count()
<< " milliseconds" << endl;
}
private:
const string name_;
const high_resolution_clock::time_point start_time_;
};
int find_using_hash(const vector<int>& input_data) {
unordered_set<int> numbers(input_data.size());
for(const auto& value : input_data) {
auto res = numbers.insert(value);
if(!res.second) {
numbers.erase(res.first);
}
}
return numbers.size() == 1 ? *numbers.begin() : -1;
}
int find_using_hashmap(const vector<int>& input_data) {
unordered_map<int,int> counter_map;
for(const auto& value : input_data) {
++counter_map[value];
}
for(const auto& map_entry : counter_map) {
if(map_entry.second % 2 == 1) {
return map_entry.first;
}
}
return -1;
}
int find_using_sort_and_count(const vector<int>& input_data) {
vector<int> local_copy(input_data);
std::sort(local_copy.begin(), local_copy.end());
int prev_value = local_copy.front();
int counter = 0;
for(const auto& value : local_copy) {
if(prev_value == value) {
++counter;
continue;
}
if(counter % 2 == 1) {
return prev_value;
}
prev_value = value;
counter = 1;
}
return counter == 1 ? prev_value : -1;
}
void execute_and_time(const string& method_name, std::function<int()> method) {
ScopedTimer timer(method_name);
cout << method_name << " returns " << method() << endl;
}
int main()
{
vector<int> input_size_vec({1<<18,1<<20,1<<22,1<<24,1<<28});
for(const auto& input_size : input_size_vec) {
// Prepare input data
std::vector<int> input_data;
const int magic_number = 123454321;
for(int i=0;i<input_size;++i) {
input_data.push_back(i);
input_data.push_back(i);
}
input_data.push_back(magic_number);
std::random_shuffle(input_data.begin(), input_data.end());
cout << "For input_size " << input_size << ":" << endl;
execute_and_time("hash-set:",std::bind(find_using_hash, input_data));
execute_and_time("sort-and-count:",std::bind(find_using_sort_and_count, input_data));
execute_and_time("hash-map:",std::bind(find_using_hashmap, input_data));
cout << "--------------------------" << endl;
}
return 0;
}
分析结果:
sh$ g++ -O3 -std=c++11 -o main *.cc
sh$ ./main
For input_size 262144:
hash-set: returns 123454321
hash-set: took 107 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 37 milliseconds
hash-map: returns 123454321
hash-map: took 109 milliseconds
--------------------------
For input_size 1048576:
hash-set: returns 123454321
hash-set: took 641 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 173 milliseconds
hash-map: returns 123454321
hash-map: took 731 milliseconds
--------------------------
For input_size 4194304:
hash-set: returns 123454321
hash-set: took 3250 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 745 milliseconds
hash-map: returns 123454321
hash-map: took 3631 milliseconds
--------------------------
For input_size 16777216:
hash-set: returns 123454321
hash-set: took 14528 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 3238 milliseconds
hash-map: returns 123454321
hash-map: took 16483 milliseconds
--------------------------
For input_size 268435456:
hash-set: returns 123454321
hash-set: took 350305 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 60396 milliseconds
hash-map: returns 123454321
hash-map: took 427841 milliseconds
--------------------------
加法
@Matt 建议的 xor 快速解决方案当然是没有竞争力的——例如，最坏情况下不到 1 秒:
int find_using_xor(const vector<int>& input_data) {
int output = 0;
for(const int& value : input_data) {
output = output^value;
}
return output;
}
For input_size 268435456:
xor: returns 123454321
xor: took 264 milliseconds
但问题仍然存在——尽管理论上算法复杂性有优势，但为什么哈希与实际排序相比效率如此低？
最佳答案
这真的取决于 hash_map/hash_set 的实现。通过将 libstdc++ 的 unordered_{map,set} 替换为 Google 的 dense_hash_{map,set}，它比 sort 快得多。 dense_hash_xxx 的缺点是它们需要两个永远不会使用的键值。有关详细信息，请参阅他们的文档。
另外要记住的是:hash_{map,set}通常会做很多动态内存分配/释放，所以最好使用libc默认的malloc/的更好的替代品免费，例如Google 的 tcmalloc 或 Facebook 的 jemalloc。
hidden $ g++ -O3 -std=c++11 xx.cpp /usr/lib/libtcmalloc_minimal.so.4
hidden $ ./a.out
For input_size 262144:
unordered-set: returns 123454321
unordered-set: took 35 milliseconds
dense-hash-set: returns 123454321
dense-hash-set: took 18 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 34 milliseconds
unordered-map: returns 123454321
unordered-map: took 36 milliseconds
dense-hash-map: returns 123454321
dense-hash-map: took 13 milliseconds
--------------------------
For input_size 1048576:
unordered-set: returns 123454321
unordered-set: took 251 milliseconds
dense-hash-set: returns 123454321
dense-hash-set: took 77 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 153 milliseconds
unordered-map: returns 123454321
unordered-map: took 220 milliseconds
dense-hash-map: returns 123454321
dense-hash-map: took 60 milliseconds
--------------------------
For input_size 4194304:
unordered-set: returns 123454321
unordered-set: took 1453 milliseconds
dense-hash-set: returns 123454321
dense-hash-set: took 357 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 596 milliseconds
unordered-map: returns 123454321
unordered-map: took 1461 milliseconds
dense-hash-map: returns 123454321
dense-hash-map: took 296 milliseconds
--------------------------
For input_size 16777216:
unordered-set: returns 123454321
unordered-set: took 6664 milliseconds
dense-hash-set: returns 123454321
dense-hash-set: took 1751 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 2513 milliseconds
unordered-map: returns 123454321
unordered-map: took 7299 milliseconds
dense-hash-map: returns 123454321
dense-hash-map: took 1364 milliseconds
--------------------------
tcmalloc: large alloc 1073741824 bytes == 0x5f392000 @
tcmalloc: large alloc 2147483648 bytes == 0x9f392000 @
tcmalloc: large alloc 4294967296 bytes == 0x11f392000 @
For input_size 268435456:
tcmalloc: large alloc 4586348544 bytes == 0x21fb92000 @
unordered-set: returns 123454321
unordered-set: took 136271 milliseconds
tcmalloc: large alloc 8589934592 bytes == 0x331974000 @
tcmalloc: large alloc 2147483648 bytes == 0x21fb92000 @
dense-hash-set: returns 123454321
dense-hash-set: took 34641 milliseconds
sort-and-count: returns 123454321
sort-and-count: took 47606 milliseconds
tcmalloc: large alloc 2443452416 bytes == 0x21fb92000 @
unordered-map: returns 123454321
unordered-map: took 176066 milliseconds
tcmalloc: large alloc 4294967296 bytes == 0x331974000 @
dense-hash-map: returns 123454321
dense-hash-map: took 26460 milliseconds
--------------------------
代码:
#include <algorithm>
#include <chrono>
#include <cmath>
#include <iostream>
#include <functional>
#include <string>
#include <vector>
#include <unordered_set>
#include <unordered_map>
#include <google/dense_hash_map>
#include <google/dense_hash_set>
using std::cout;
using std::chrono::high_resolution_clock;
using std::chrono::milliseconds;
using std::endl;
using std::string;
using std::vector;
using std::unordered_map;
using std::unordered_set;
using google::dense_hash_map;
using google::dense_hash_set;
class ScopedTimer {
public:
ScopedTimer(const string& name)
: name_(name), start_time_(high_resolution_clock::now()) {}
~ScopedTimer() {
cout << name_ << " took "
<< std::chrono::duration_cast<milliseconds>(
high_resolution_clock::now() - start_time_).count()
<< " milliseconds" << endl;
}
private:
const string name_;
const high_resolution_clock::time_point start_time_;
};
int find_using_unordered_set(const vector<int>& input_data) {
unordered_set<int> numbers(input_data.size());
for(const auto& value : input_data) {
auto res = numbers.insert(value);
if(!res.second) {
numbers.erase(res.first);
}
}
return numbers.size() == 1 ? *numbers.begin() : -1;
}
int find_using_unordered_map(const vector<int>& input_data) {
unordered_map<int,int> counter_map;
for(const auto& value : input_data) {
++counter_map[value];
}
for(const auto& map_entry : counter_map) {
if(map_entry.second % 2 == 1) {
return map_entry.first;
}
}
return -1;
}
int find_using_dense_hash_set(const vector<int>& input_data) {
dense_hash_set<int> numbers(input_data.size());
numbers.set_deleted_key(-1);
numbers.set_empty_key(-2);
for(const auto& value : input_data) {
auto res = numbers.insert(value);
if(!res.second) {
numbers.erase(res.first);
}
}
return numbers.size() == 1 ? *numbers.begin() : -1;
}
int find_using_dense_hash_map(const vector<int>& input_data) {
dense_hash_map<int,int> counter_map;
counter_map.set_deleted_key(-1);
counter_map.set_empty_key(-2);
for(const auto& value : input_data) {
++counter_map[value];
}
for(const auto& map_entry : counter_map) {
if(map_entry.second % 2 == 1) {
return map_entry.first;
}
}
return -1;
}
int find_using_sort_and_count(const vector<int>& input_data) {
vector<int> local_copy(input_data);
std::sort(local_copy.begin(), local_copy.end());
int prev_value = local_copy.front();
int counter = 0;
for(const auto& value : local_copy) {
if(prev_value == value) {
++counter;
continue;
}
if(counter % 2 == 1) {
return prev_value;
}
prev_value = value;
counter = 1;
}
return counter == 1 ? prev_value : -1;
}
void execute_and_time(const string& method_name, std::function<int()> method) {
ScopedTimer timer(method_name);
cout << method_name << " returns " << method() << endl;
}
int main()
{
vector<int> input_size_vec({1<<18,1<<20,1<<22,1<<24,1<<28});
for(const auto& input_size : input_size_vec) {
// Prepare input data
std::vector<int> input_data;
const int magic_number = 123454321;
for(int i=0;i<input_size;++i) {
input_data.push_back(i);
input_data.push_back(i);
}
input_data.push_back(magic_number);
std::random_shuffle(input_data.begin(), input_data.end());
cout << "For input_size " << input_size << ":" << endl;
execute_and_time("unordered-set:",std::bind(find_using_unordered_set, std::cref(input_data)));
execute_and_time("dense-hash-set:",std::bind(find_using_dense_hash_set, std::cref(input_data)));
execute_and_time("sort-and-count:",std::bind(find_using_sort_and_count, std::cref(input_data)));
execute_and_time("unordered-map:",std::bind(find_using_unordered_map, std::cref(input_data)));
execute_and_time("dense-hash-map:",std::bind(find_using_dense_hash_map, std::cref(input_data)));
cout << "--------------------------" << endl;
}
return 0;
}
关于c++ - O(N) 算法比 O(N logN) 算法慢，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/27932832/
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