最长上升子序列

Author: EntropyIncreaser

longest_increasing_sequence.md

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+# 问题简介
+最长公共上升子序列，longest increasing sequence，简称LCIS
+# 思路
+这个问题要由动态规划来解决，时间复杂度是O(n<sup>2</sup>)，建立dp[n]和track[n]    
+意义：
+----
+* dp[i]：以数据a[i]结尾的最长上升序列长度
+* track[i]：数据a[i]作为最长上升序列末项的前一项索引
+
+

longest_increasing_sequence.py

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+#!/usr/bin/python
+#coding: utf-8
+
+def isBiggerCompare(a, b):
+	return a > b
+
+def findLIS(sequence, compare = isBiggerCompare):
+	n = len(sequence)
+	dp = [0 for i in range(n)]
+	track = [-1 for i in range(n)]
+	ans = 1
+	for i in range(1, n):
+		MAX = 0
+		for j in range(i):
+			if compare(sequence[i], sequence[j]) and MAX < dp[j]:
+				MAX = dp[j]
+				track[i] = j
+				# track[i] sequence[i]作为最长上升序列末项的前一项
+		dp[i] = MAX + 1
+		# dp[i] 以sequence[i]结尾的最长上升序列长度
+		if dp[i] > dp[ans]:
+			ans = i
+	ansList = [sequence[ans]]
+	while track[ans] != -1:
+		ans = track[ans]
+		ansList.insert(0, sequence[ans])
+	return ansList
+
+if __name__ == '__main__':
+	s = [3, 1, 2, 1, 4, 3, 5]
+	print findLIS(s)