92 lines
3.8 KiB
Text
92 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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- stacks-queues
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- intro-ordered
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description: "Moving a subarray across an array to efficently compute information."
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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 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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### Implementation
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<IncompleteSection />
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## Sliding Window
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<Problems 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 an ordered 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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### Implementation
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<IncompleteSection />
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### Problems
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<Problems 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 problems={metadata.problems.qs} />
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