一、问题描述
输入一组整数，求出这组数字子序列和中最大值。也就是只要求出最大子序列的和，不必求出最大的那个序列。例如：
- 序列：-2 11 -4 13 -5 -2，则最大子序列和为20。
- 序列：-6 2 4 -7 5 3 2 -1 6 -9 10 -2，则最大子序列和为16。
二、PHP代码实现
实现可参考：http://www.cnblogs.com/CCBB/archive/2009/04/25/1443455.html
<?php
/**
* 找出最大子序列和
*/
// 方法一、 最暴力的方法 O(N^3)
function findMax1($data){
$max_value = 0;
$max_start = 0;
$max_end = 0;
$data_num = count($data);
for($i=0;$i<$data_num;$i++){
for($j=$i;$j<$data_num;$j++){
$sum = 0;
$temp = array();
for($k=$i;$k<=$j;$k++){
$temp[] = $data[$k];
$sum += $data[$k];
}
if($max_value < $sum){
$max_value = $sum;
$max_start = $i;
$max_end = $j;
}
}
}
return array($max_value,$max_start,$max_end);
}
//方法二、 记录上一次的结果 O(N^2)
function findMax2($data){
$max_value = 0;
$max_start = 0;
$max_end = 0;
$data_num = count($data);
for($i=0;$i<$data_num;$i++){
$last_sum = 0;
for($j=$i;$j<$data_num;$j++){
$sum = $last_sum+$data[$j];
if($max_value < $sum){
$max_value = $sum;
$max_start = $i;
$max_end = $j;
}
$last_sum = $sum;
}
}
return array($max_value,$max_start,$max_end);
}
//方法三、分而自治算法 O(NlogN)
function findMax3($data,$start=0,$end=0){
if($start >= $end){
$max_value = isset($data[$start])?$data[$start]:0;
}else{
$mid = floor(($start+$end)/2);
list($max_left_value,$max_left_start,$max_left_end) = findMax3($data,$start,$mid);
list($max_right_value,$max_right_start,$max_right_end) = findMax3($data,$mid+1,$end);
//计算左边的最大值
$max_this_value = $data[$mid];
$max_this_start = $mid;
$max_this_end = $mid;
$temp = 0;
for($i=$max_this_start;$i>=$start;$i--){
$temp += $data[$i];
if($temp > $max_this_value){
$max_this_value = $temp;
$max_this_start = $i;
}
}
//计算右边的最大值
$temp = $max_this_value;
for($i=$max_this_end+1;$i<=$end;$i++){
$temp += $data[$i];
if($temp > $max_this_value){
$max_this_value = $temp;
$max_this_end = $i;
}
}
$max_value = $max_this_value;
$start = $max_this_start;
$end = $max_this_end;
if($max_value < $max_left_value){
$max_value = $max_left_value;
$start = $max_left_start;
$end = $max_left_end;
}
if($max_value < $max_right_value){
$max_value = $max_right_value;
$start = $max_right_start;
$end = $max_right_end;
}
}
return array($max_value,$start,$end);
}
//方法四、找规律算法 O(N)
function findMax4($data){
$max_value = 0;
$thisSum = 0;
$data_len = count($data);
$start = 0;
$end = 0;
for($i=0;$i<$data_len;$i++){
$thisSum += $data[$i];
if($thisSum > $max_value){
$max_value = $thisSum;
$end =$i;
}else if($thisSum < 0){
$thisSum = 0;
$start= $i+1;
}
}
return array($max_value,$start,$end);
}
三、测试
<?php
// 获取一个随机数列
function getArrayData($len=0){
$arr_data = array('4','-6','8','2','-4','7','2','-5','1');
if($len != 0){
$arr_data = array();
for($i=0;$i<$len;$i++){
$arr_data[] = rand(-9,9);
}
}
return $arr_data;
}
$arr_data = getArrayData(1000);
echo implode(',',$arr_data);
echo "\n";
echo "*************findMax1:最暴力的方法**********\n";
$start_time = time();
list($max_value1,$start1,$end1)=findMax1($arr_data);
echo "max_value:{$max_value1}\n";
echo "index:{$start1}-{$end1}\n";
echo 'take time:'.(time()-$start_time);echo "\n";
echo "*************findMax2:记录上一次的结果**********\n";
$start_time = time();
list($max_value2,$start2,$end2)=findMax2($arr_data);
echo "max_value:{$max_value2}\n";
echo "index:{$start2}-{$end2}\n";
echo 'take time:'.(time()-$start_time);echo "\n";
echo "*************findMax3:分而治之算法**********\n";
$start_time = time();
list($max_value3,$start3,$end3)=findMax3($arr_data,0,count($arr_data)-1);
echo "max_value:{$max_value3}\n";
echo "index:{$start3}-{$end3}";echo "\n";
echo 'take time:'.(time()-$start_time);echo "\n";
echo "*************findMax4:找规律算法**********\n";
$start_time = time();
list($max_value4,$start4,$end4)=findMax4($arr_data);
echo "max_value:{$max_value4}\n";
echo "{$start4}-{$end4}";echo "\n";
echo 'take time:'.(time()-$start_time);echo "\n";
四、测试结果
-
n=1000
测试结果:n=1000 - n=5000
还在测试中，第一种方案太慢了
