分治算法

一.總述

分治算法其實就是將一個大問題分解爲若干個類型相同可是規模較小的子問題，使用遞歸的方式一直分解下去，而後將子問題的解合併獲得原問題的解的策略。

二.經典的分治算法列舉

二分搜索、大整數乘法、strassen矩陣乘法、棋盤覆蓋、合併排序、快速排序、線性時間選擇、最接近點對問題、循環賽日程表、漢諾塔等

三.分治算法舉例

最大子數列問題：下面是股票的價格變更，求在哪一天買入，哪一天賣出得到的收益最高

 

 

 1.雙重循環實現（不是分治算法）

1）C#實現及結果

            int[] prices = { 100, 113, 110, 85, 105, 102, 86, 63, 81, 101, 94, 106, 101, 79, 94, 90, 97 };

            int profit = prices[1] - prices[0];
            int tempProfit = 0;
            int[] result = { 0, 1 };

            for(int i = 0;i < prices.Length;i++)
            {
                for(int j = i;j < prices.Length; j++)
                {
                    tempProfit = prices[j] - prices[i];
                    if(profit < tempProfit)
                    {
                        profit = tempProfit;
                        result[0] = i;
                        result[1] = j;
                    }
                }
            }

            Console.WriteLine("在第" + result[0] + "天買入，第" + result[1] + "天賣出，所得到利潤最多，每股利潤是" + profit + "元");

 

 2）Lua實現代碼及結果

prices = { 100, 113, 110, 85, 105, 102, 86, 63, 81, 101, 94, 106, 101, 79, 94, 90, 97 }
profit = prices[2] - prices[1]
tempProfit = 0
result = { 0, 1 }

for i=1,#prices,1 do
    for j=i,#prices,1 do
        tempProfit = prices[j] - prices[i];
        if (profit < tempProfit) then
            profit = tempProfit
            result[0] = i-1
            result[1] = j-1
        end
    end
end

print("在第"..result[0].."天買入，第"..result[1].."天賣出，所得到利潤最多，每股利潤是"..profit.."元")

 

 2.分治算法

1）c#實現

static void Main(string[] args)
        {
            int[] prices = { 100, 113, 110, 85, 105, 102, 86, 63, 81, 101, 94, 106, 101, 79, 94, 90, 97 };

            int start = 0;
            int end = prices.Length - 1;
            int profit = GetMaxProfit(prices,ref start,ref end);

            Console.WriteLine("在第" + start + "天買入，第" + end + "天賣出，所得到利潤最多，每股利潤是" + profit + "元");
        }

        static int MaxNumInArray(int[] nums,int start,int end,out int maxNumIndex)
        {
            int result = nums[start];
            maxNumIndex = start;
            for(int i = start +1;i <= end; i++)
            {
                if (nums[i] > result)
                {
                    result = nums[i];
                    maxNumIndex = i;
                }
            }
            return result;
        }

        static int MinNumInArray(int[] nums,int start,int end,out int minNumIndex)
        {
            int result = nums[start];
            minNumIndex = start;
            for (int i = start + 1; i <= end; i++)
            {
                if (nums[i] < result) 
                {
                    result = nums[i];
                    minNumIndex = i;
                }
            }
            return result;
        }

        static int GetMaxProfit(int[] nums,ref int start,ref int end)
        {
            if (start == end)
                return 0;

            int mid = (start + end) / 2;

            int crossStart;
            int crossEnd;
            int crossProfit = MaxNumInArray(nums,mid + 1,end,out crossEnd) - MinNumInArray(nums,start,mid,out crossStart);
            int leftStart = start;
            int leftEnd = mid;
            int leftProfit = GetMaxProfit(nums,ref leftStart,ref leftEnd);
            int rightStart = mid + 1;
            int rightEnd = end;
            int rightProfit = GetMaxProfit(nums,ref rightStart,ref rightEnd);

            int result;
            if(crossProfit > leftProfit && crossProfit > rightProfit)
            {
                result = crossProfit;
                start = crossStart;
                end = crossEnd;
            }
            else if(leftProfit > crossProfit && leftProfit > rightProfit)
            {
                result = leftProfit;
                start = leftStart;
                end = leftEnd;
            }
            else
            {
                result = rightProfit;
                start = rightStart;
                end = rightEnd;
            }

            return result;
        }

 

 2）.Lua實現

prices = { 100, 113, 110, 85, 105, 102, 86, 63, 81, 101, 94, 106, 101, 79, 94, 90, 97 }

function MaxNumInArray(nums,smax,emax)
    resultmax = nums[smax]
    maxNumIndex = smax
    for imax = smax,emax,1 do
        if (nums[imax] > resultmax) then
            resultmax = nums[imax]
            maxNumIndex = imax
        end
    end
    return resultmax,maxNumIndex;
end

function MinNumInArray(nums,smin,emin)
    resultmin = nums[smin]
    minNumIndex = smin
    for i = smin +1,emin,1 do
        if (nums[i] < resultmin) then
            resultmin = nums[i]
            minNumIndex = i
        end
    end
    return resultmin,minNumIndex
end

function GetMaxProfit(nums,s,e)
    if s==e then
        return 1,s,e
    end

    local mid = 0;
    if (s + e) % 2 == 0 then
        mid = (s + e) / 2
    else
        mid = (s + e - 1) / 2
    end

    local rightMax,crossEnd = MaxNumInArray(nums,mid + 1,e)
    local leftMin,crossStart = MinNumInArray(nums,s,mid)
    local crossProfit = rightMax - leftMin
    local leftProfit,leftStart,leftEnd = GetMaxProfit(nums,s,mid)
    local rightProfit,rightStart,rightEnd = GetMaxProfit(nums,mid+1,e)

    if crossProfit > leftProfit and crossProfit > rightProfit then
        result = crossProfit
        s = crossStart
        e = crossEnd
    elseif (leftProfit > crossProfit and leftProfit > rightProfit) then
        result = leftProfit
        s = leftStart
        e = leftEnd
    else
        result = rightProfit;
        s = rightStart;
        e = rightEnd;
    end

    return result,s,e
end

profit,s,e = GetMaxProfit(prices,1,#prices)

print("在第"..(s-1).."天買入，第"..(e-1).."天賣出，所得到利潤最多，每股利潤是"..profit.."元")