All Tracks Algorithms Searching Binary Search Problem

Two Arrays
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Algorithms, Binary search algorithm, Prefix sum, Searching algorithm, Two pointer, prefix-sum

Problem
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Let's define the matching number for two arrays X and Y, each with size n, as the number of pairs of indices \((i,j)\) such that \(X_i = Y_j\) (\(1 \le i,j \le n\)).

You are given an integer K and two arrays A and B with sizes N and M respectively. Find the number of ways to pick two subarrays with equal size, one from array A and the other from array B, such that the matching number for these two subarrays is \(\ge K\).

Note: a subarray consists of one or more consecutive elements of an array.

Constraints

  • \( 1 \le N , M \le 2,000 \)

  • \( 1 \le A_i \le 1,000,000,000 \) for each valid i

  • \( 1 \le B_i \le 1,000,000,000 \) for each valid i

  • \( 1 \le K \le NM \)

Input format

The first line of the input contains three space-separated integers N, M and K.

The second line contains N space-separated integers \(A_1, \dots, A_N\).

The third line contains M space-separated integers \( B_1, \dots, B_M \).

Output Format

Print a single line containing one number denoting the number of ways to choose the pair of subarrays.

SAMPLE INPUT
3 3 1
1 2 3
1 2 3
SAMPLE OUTPUT
8
Explanation

The following pairs of subarrays have matching number \(\ge 1\): ([1], [1]), ([2], [2]), ([3], [3]), ([1, 2], [1, 2]), ([1, 2], [2, 3]), ([2, 3], [1, 2]), ([2, 3], [2, 3]), ([1, 2, 3], [1, 2, 3]).

Time Limit: 1.0 sec(s) for each input file.
Memory Limit: 256 MB
Source Limit: 1024 KB

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