**Problem:**

Please find the problem here.

**Solution:**

Competitive programming hinted on this is a problem using Segment Tree, but reflecting on the problem it look much more like an Interval tree. The basic idea is that the numbers themselves are really meaningless, we only care about the frequency value, so we keep the ranges of identical numbers like this

-1 - 1 1 1 1 1 3 10 10 10

becomes

[0, 2) [2, 6) [6, 7), [7, 10)

Then we build the segment tree bottom up. That gives

[0, 10) has answer 4 - left-> [0, 6) has answer 4 - left-> [0, 2) has answer 2

-right-> [2, 6) has answer 4

-right-> [6, 10) has answer 3 - left-> [6, 7) has answer 1

-right-> [7, 10) has answer 3

Building this tree bottom up is simple, we process it layer by layer.

First, we scan the ranges in the bottom layer, once we get two ranges, we built its parent node. If there is an orphan range just promote it to the parent layer.

Continue to build layers until there is only one node left, that complete the tree building, the time is O(n log n).

The trouble of building the tree pays off when we do the queries. When we query, if a range matches a tree node exactly, gives the answer! Otherwise if the range is a leaf node, just answer the length of the range itself, otherwise recursively compute the answer of the left and right sub-tree and take the maximum.

The interval tree technique used above is probably not optimal as the book mention there is a O(n) dynamic programming solution. Looking forward to that. The approach seems to generalize to whatever problem that involve intervals and combining answer from interval is easy.

There is a fun story associated with the solving of this problem. I have submitted the solution 7 times to the online judge and still getting wrong answer, turn out I had 2 bug, the first bug is that the input is an inclusive interval with 1 based index and my representation is 0 based index open close intervals, and the second bug is really bad - I missed the 1st line to say the input contains multiple test cases. Again, I coded an oracle using brute force algorithm and found the first bug.

What I learnt?

- I have gone through some literature about segment tree.
- I thought about how to generalize this problem solving paradigm to other problems related to intervals.
- Read the problem.
- Code an oracle and test with random inputs is good strategy.

**Code:**

#include "stdafx.h" // http://uva.onlinejudge.org/index.php?option=onlinejudge&page=show_problem&problem=2176 #include "UVa11235.h" #include <iostream> #include <list> #include <map> #include <algorithm> using namespace std; struct interval_tree_node { int from; int to; int answer; /* the frequency of the most frequent number in the interval */ interval_tree_node* left; interval_tree_node* right; interval_tree_node(int from, int to, int answer) : from(from), to(to), answer(answer), left(NULL), right(NULL) { } ~interval_tree_node() { if (this->left != NULL) { delete this->left; } if (this->right != NULL) { delete this->right; } } }; int process_query(interval_tree_node* node, int query_from, int query_to) { if (node->from == query_from && node->to == query_to) { return node->answer; } else { if (node->left == NULL) { // I am at a leaf node now - the best answer is just the input range return query_to - query_from; } else { int leftFrom = max(node->left->from, query_from); int leftTo = min(node->left->to, query_to); int rightFrom = max(node->right->from, query_from); int rightTo = min(node->right->to, query_to); int left_answer = leftFrom < leftTo ? process_query(node->left, leftFrom, leftTo) : 0; int right_answer = rightFrom < rightTo ? process_query(node->right, rightFrom, rightTo) : 0; return max(left_answer, right_answer); } } } int UVa11235() { while (true) { int number_of_values, number_of_queries; cin >> number_of_values; if (number_of_values == 0) { break; } cin >> number_of_queries; list<interval_tree_node*> ranges; // Step 1: Read the input as ranges bool first = true; int last = -1; int last_index = 0; for (int i = 0; i < number_of_values; i++) { int current_value; cin >> current_value; if (first) { first = false; } else { if (last != current_value) { ranges.push_back(new interval_tree_node(last_index, i, (i - last_index))); last_index = i; } } last = current_value; } ranges.push_back(new interval_tree_node(last_index, number_of_values, (number_of_values - last_index))); // Step 2: Build the tree list<interval_tree_node*> empty_list; list<interval_tree_node*>* layer_below = &ranges; list<interval_tree_node*>* layer_above = &empty_list; while (layer_below->size() > 1) { interval_tree_node* left = NULL; interval_tree_node* right = NULL; for (list<interval_tree_node*>::iterator ii = layer_below->begin(); ii != layer_below->end(); ii++) { if (left == NULL) { left = *ii; } else { // Step 2.1: Any two nodes in the bottom layer has a parent node right = *ii; interval_tree_node* parent = new interval_tree_node(left->from, right->to, max(left->answer, right->answer)); parent->left = left; parent->right = right; left = right = NULL; layer_above->push_back(parent); } } if (left != NULL) { // Step 2.2: The orphan last node will be made to the parent layer layer_above->push_back(left); } // Step 2.3: Swap the role of above and below to reuse the list list<interval_tree_node*>* temp = layer_below; layer_below = layer_above; layer_above = temp; // Step 2.4: Prepare for the next iteration, above should be empty layer_above->clear(); } interval_tree_node* root = *(layer_below->begin()); for (int i = 0; i < number_of_queries; i++) { int query_from; int query_to; cin >> query_from; cin >> query_to; /* * query_from - 1 is conversion from 1 based index to 0 based, * query_to is conversion from 1 based index to 0 based, and then become an exclusive index */ cout << process_query(root, query_from - 1, query_to) << endl; } delete root; } return 0; }

thanks :)

ReplyDeletethanks :)

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