From 4d9134a71580cd613febb518911fb0ac5e42e3d5 Mon Sep 17 00:00:00 2001 From: "phluenam@gmail.com" Date: Wed, 20 Sep 2023 01:17:03 +0800 Subject: [PATCH] Add 1122 --- md/1122.md | 39 +++++++++++++++++++++++++++++++++++++++ media/1122/1.png | Bin 0 -> 230437 bytes media/1122/2.png | Bin 0 -> 229474 bytes 3 files changed, 39 insertions(+) create mode 100644 md/1122.md create mode 100644 media/1122/1.png create mode 100644 media/1122/2.png diff --git a/md/1122.md b/md/1122.md new file mode 100644 index 0000000..f31c79c --- /dev/null +++ b/md/1122.md @@ -0,0 +1,39 @@ +หากแปลโจทย์ข้อนี้เป็นภาษา Graph Theory จะได้คำถามว่าหากมี Node $N$ ตัว (แทนนักเรียน) และ Edge $M$ อัน(แทนความเป็นเพื่อน) จะมีเซ็ตของ Node 4 ตัวกี่เซ็ตที่ใน Subgraph ของ Node 4 ตัวนั้นมี Edge อย่างน้อย 5 อัน + +วิธีคิดสำหรับข้อนี้คือพิจารณา Subgraph ขนาด 4 ที่มี 5 หรือ 6 Edge (4 Node จะมีได้อย่างมาก 6 Edge ใน Simple Graph) + +## Adjacency Matrix + +ในข้อนี้เราจะต้องการเช็คว่า Node $a$ และ $b$ มี Edge ที่เชื่อมต่อกันไหมในเวลา $\mathcal{O}(1)$ + +โครงสร้างข้อมูลที่เหมาะสมคือ Adjacency Matrix ซึ่งเป็นเป็น Array สองมิติ $A$ ขนาด $N \times N$ โดย $A[a][b]$ จะเป็น $1$ หากมี Edge ดังกล่าวและ $0$ หากไม่มี + +การสร้าง Matrix นี้ใช้เวลาและพื้นที่ความจำ $\mathcal{O}(N^2)$ (เพราะต้องสร้าง Memory ขนาด $N \times N$) + +การอ่านแต่ละ Edge และตั้งค่า $A[X_i][Y_i]$ ให้เป็น $1$ ใช้เวลา $\mathcal{O}(M)$ + +## Solution + +### เคส 6 Edge + +![](../media/1122/1.png) + +สังเกตว่าเราสำหรับ Subgraph ที่เข้าเคส 6 Edge (ซึ่งเป็น Complete Graph) จะต้องมีคู่ Edge ที่ไม่มี Node ร่วมกัน 3 คู่ + +ดังนั้นเมื่อพิจารณาคู่ Edge $((X_i,Y_i), (X_j,Y_j))$ คู่หนึ่งจาก $M$ Edge ที่ได้มาจากข้อมูลนำเข้า โดยทั้งสอง Edge ไม่ได้มี Node ร่วมกัน ใน 4 Node $X_i, Y_i, X_j, Y_j$ หากใน 4 Node นี้มี Edge เชื่อมกันครบ 6 อัน จะสามารถนับว่าเจอ Subgraph เคส 6 Edge ได้ (โดยนับจำนวน Edge ได้โดยตรงด้วยการบวก $A[X_i][Y_i] + A[X_i][X_j] + A[X_i][Y_j] + A[Y_i][X_j] + A[Y_i][Y_j] + A[X_j][Y_j]$ จาก Adjacency Matrix ที่เก็บไว้) + +แต่เมื่อนับเช่นนี้จะเกิดการนับซ้ำ 3 รอบ เนื่องจากในแต่ละ Graph ที่เข้าเคสนี้จะมีคู่ Edge ที่ไม่มี Node ร่วมกัน 3 คู่ จึงต้องนำจำนวนที่นับได้มาหาร 3 เพื่อให้ได้จำนวนเคส 6 Edge ที่ต้องการ + +### เคส 5 Edge + +![](../media/1122/1.png) + +สำหรับ Subgraph ที่เข้าเคส 5 Edge สังเกตว่าสามารถมองว่าเกิดจากการลบ 1 Edge จากเคส 6 Edge ด้านบน จะเหลือคู่ Edge ที่ไม่มี Node ร่วมกัน 2 คู่ + +เมื่อพิจารณาคู่ Edge $((X_i,Y_i), (X_j,Y_j))$ คู่หนึ่งที่ไม่มี Node ร่วมกันดังเช่นในเคสก่อนหน้า หากใน 4 Node นี้มี Edge เชื่อมกัน 5 อันพอดี จะสามารถนับว่าเจอ Subgraph เคส 5 Edge ได้ แต่ต้องหาร 2 เพราะจะถูกนับซ้ำสำหรับทั้ง 2 คู่ + +คำตอบที่ต้องการคือนำจำนวนที่นับได้จากทั้งสองเคสมาบวกกัน + + +เนื่องจากมี $M$ Edge จะทำให้มี $\mathcal{O}(M^2)$ คู่ของ Edge ที่ต้องพิจารณา การนับ Edge ใน Subgraph 4 Node ใช้เวลา $\mathcal{O}(1)$ ดังนั้นทั้งปัญหาจึงใช้เวลา $\mathcal{O}(N^2+M^2)$ + diff --git a/media/1122/1.png b/media/1122/1.png new file mode 100644 index 0000000000000000000000000000000000000000..8515633ff69e401abd721be0ee35471b6b00dc68 GIT binary patch literal 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