From 1609a58c3c1f3858f8d67a1c8c861eac1174d0a1 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Thomas=20Vo=C3=9F?= <thomas.voss.bochum@gmail.com>
Date: Tue, 7 Jul 2015 10:39:57 +0200
Subject: [PATCH] Rework example 7 and provide an alternative way of solving
 the problem.

---
 c++/Chapter 11/Chapter11_7.cpp | 104 ++++++++++++++++++++++-----------
 1 file changed, 71 insertions(+), 33 deletions(-)

diff --git a/c++/Chapter 11/Chapter11_7.cpp b/c++/Chapter 11/Chapter11_7.cpp
index f7aea593..2dccb4db 100644
--- a/c++/Chapter 11/Chapter11_7.cpp	
+++ b/c++/Chapter 11/Chapter11_7.cpp	
@@ -1,51 +1,89 @@
+#include <algorithm>
 #include <iostream>
+#include <stack>
+#include <tuple>
 #include <vector>
+
 using namespace std;
 
-struct htWt {
+struct Person {
 	int height, weight;
 };
 
-void sortHeight(htWt arr[], int n) {
-	for (int i=0; i<n; i++) {
-		for (int j=0; j<n-1; j++){
-			if (arr[j].height>arr[j+1].height) {
-				htWt tmp = arr[j];
-				arr[j] = arr[j+1];
-				arr[j+1] = tmp;
-			}
+bool operator<(const Person& lhs, const Person& rhs) {
+	return std::tie(lhs.height, lhs.weight) < std::tie(rhs.height, rhs.weight);
+}
+
+// Time complexity: O(n + n * log(n)) = O(n * log(n))
+// Space complexity: O(1)
+template<typename T>
+unsigned int CalculateTower(std::vector<T>& persons) {
+	make_heap(persons.begin(), persons.end());
+	auto rit = persons.end();
 
+	unsigned int result = 0;
+	Person current{std::numeric_limits<int>::max(), std::numeric_limits<int>::max()};
+
+	while (rit >= persons.begin()) {
+		if (persons.front() < current) {
+			current = persons.front();
+			result++;
 		}
+
+		pop_heap(persons.begin(), rit);
+		rit = prev(rit);
 	}
+
+	return result;
 }
 
-int getIncreasingSequence(htWt arr[], int n) {
-	sortHeight(arr, n);
-	vector<vector<int> > wt;
-	for (int i=0; i<n; i++) {
-		bool flag = true;
-		for (int j=0; j<wt.size(); j++) {
-			if (arr[i].weight >= wt[j].back()) {
-				flag = false;
-				wt[j].push_back(arr[i].weight);
+// Time complexity: O(n^2)
+// Space complexity: O(n)
+template<typename T, typename Comparator = std::less<T>>
+unsigned int LongestIncreasingSubsequence(const std::vector<T>& values, Comparator comp = Comparator()) {
+	vector<unsigned int> l(values.size(), 1);
+	unsigned int best = 0;
+	for (int i = 1; i < values.size(); i++) {
+		unsigned int max = 0;
+		for (int j = i - 1; j >= 0; j--) {
+			if (comp(values[j], values[i])) {
+				max = std::max(max, l[j]);
 			}
 		}
-		if (flag) {
-			vector<int> v;
-			v.push_back(arr[i].weight);
-			wt.push_back(v);
-		}
+		l[i] = ++max;
+		best = std::max(best, max);
 	}
-	int max = 0;
-	for (int i=0; i<wt.size(); i++) {
-		if (max < wt[i].size())
-			max = wt[i].size();
-	}
-	return max;
+
+	return best;
 }
 
-int main(){
-	htWt arr[] = {{12, 13}, {11, 15}, {9, 20}, {20, 20}, {40, 21}, {8, 42}};
-	cout<<getIncreasingSequence(arr, 6);
+int main() {
+	vector<Person> persons{{12, 13}, {11, 15}, {9, 20}, {20, 20}, {40, 21}, {8, 42}};
+
+	// We calculate the maximum height of a tower of persons by relying on a max-heap,
+	// popping persons of the heap onto the tower if the current max element in the heap
+	// is strictly smaller than the current top-most person in the tower. If it is not strictly
+	// smaller, we just drop the person as they are "as good as"/"incomparable to" the person
+	// already standing on top of the tower. Please note that this works because we are not maximizing
+	// the total height of the tower, but only the number of people being involved in it.
+	{
+		vector<Person> cpersons = persons;
+		cout << "With heap: " << CalculateTower(cpersons) << "\n";
+	}
+	// We are not altering the original sequence as in the original problem statement for
+	// the longest increasing subsequence problem. For the example setup presented here,
+	// the longest strictly increasing subsequence is 3.
+	{
+		cout << "Without sorting: " << LongestIncreasingSubsequence(persons) << "\n";
+	}
+	// We are mutating the original sequence of values here, requiring O(n log(n)) to do so.
+	// Alternatively, we could just rely on a heap to achieve the same functionality.
+	{
+		vector<Person> cpersons(persons);
+		sort(cpersons.begin(), cpersons.end(), [](const Person& lhs, const Person& rhs) {
+			return lhs.height < rhs.height;
+		});
+		cout << "With sorting by height: " << LongestIncreasingSubsequence(cpersons) << "\n";
+	}
 	return 0;
-}
\ No newline at end of file
+}