83 lines
3.8 KiB
Text
83 lines
3.8 KiB
Text
---
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id: sliding
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title: "Sliding Window"
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author: Darren Yao?, Benjamin Qi
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prerequisites:
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- Silver - Stacks & Queues
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- Silver - Introduction to Ordered Maps & Sets
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description: "?"
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frequency: 2
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---
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import { Problem } from "../models";
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export const metadata = {
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problems: {
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sample: [
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new Problem("CSES", "Sum of Two Values", "1141", "Easy", false, []),
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],
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slide: [
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new Problem("CSES", "Playlist", "1141", "Easy", false, [],"classic example of 2P"),
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],
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general: [
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new Problem("CSES", "Subarray Sums I", "1660", "Easy", false, [], ""),
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new Problem("CSES", "Sliding Median", "1076", "Easy", false, [], ""),
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new Problem("CSES", "Sliding Cost", "1077", "Hard", false, [], ""),
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new Problem("CSES", "Max Subarray Sum II", "1644", "Normal", false, [], ""),
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new Problem("Silver", "Diamond Collector", "643", "Easy", false, ["2P", "Sorting"], ""),
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new Problem("Silver", "Paired Up", "738", "Normal", false, ["2P", "Sorting"]),
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new Problem("Gold", "Haybale Feast", "767", "Normal", false, ["Set", "Sliding Window"]),
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new Problem("CF", "Books", "problemset/problem/279/B", "Normal", false, []),
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new Problem("CF", "Cellular Network", "problemset/problem/702/C", "Normal", false, []),
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new Problem("CF", "USB vs. PS/2", "problemset/problem/762/B", "Normal", false, []),
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new Problem("CF", "K-Good Segment", "problemset/problem/616/D", "Normal", false, []),
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new Problem("CF", "Garland", "problemset/problem/814/C", "Normal", false, []),
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new Problem("CF", "Jury Meeting", "problemset/problem/853/B", "Normal", false, []),
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new Problem("Plat", "Fort Moo", "600", "Very Hard", false, ["Sliding Window"]),
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],
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qs: [
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new Problem("YS","Queue Composite","queue_operate_all_composite","Hard",true,[],""),
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],
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}
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};
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## Two Pointers
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<problems-list problems={metadata.problems.sample} />
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Two pointers refers to iterating two monotonic pointers across an array to search for a pair of indices satisfying some condition in linear time.
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<resources>
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<resource source="CPH" title="8.1 - Two Pointers"></resource>
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<resource source="IUSACO" title="14.1 - Two Pointers"></resource>
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</resources>
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## Sliding Window
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<problems-list problems={metadata.problems.slide} />
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Let's envision a **sliding window** (or constant size subarray) of size $K$ moving left to right along an array, $a$.
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For each position of the window, we want to compute some information. For example, we could store a `std::set` of integers representing the integers inside the window. If the window currently spans the range $i \dots j$, we observe that moving the range forward to $i+1 \dots j+1$ only removes $a_i$ and adds $a_{j+1}$ to the window. We can support these two operations and query for the minimum / maximum in the set in $O(\log N)$.
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To compute the sum in the range, instead of using a set, we can store a variable $s$ representing the sum. As we move the window forward, we update $s$ by subtracting $a_i$ from $s$ and adding $a_{j+1}$ to $s$.
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<resources>
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<resource source="Medium" title="Introduction to Sliding Window Algorithms" url="https://levelup.gitconnected.com/an-introduction-to-sliding-window-algorithms-5533c4fe1cc7"> </resource>
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<resource source="GFG" title="Window Sliding Technique" url="window-sliding-technique"> </resource>
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</resources>
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### Problems
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<problems-list problems={metadata.problems.general} />
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## Sliding Window Minimum in $O(N)$
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<resources>
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<resource source="cp-algo" title="Minimum stack / Minimum queue" url="data_structures/stack_queue_modification.html" starred>Mentions two ways to solve this (both are important)!</resource>
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</resources>
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In particular, the second method allows us to solve the following generalization in linear time as well:
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<problems-list problems={metadata.problems.qs} />
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