From 8d9a5c28678645852f2545c3df8938e70fbb44f6 Mon Sep 17 00:00:00 2001 From: Phluenam Date: Mon, 30 Oct 2023 00:52:05 +0800 Subject: [PATCH] Add 1061 (#150) * Add 1061 * fix img path --------- Co-authored-by: Borworntat Dendumrongkul <64594596+MasterIceZ@users.noreply.github.com> --- md/1061.md | 121 +++++++++++++++++++++++++++++++++++++++++++++++ media/1061/0.png | Bin 0 -> 11079 bytes media/1061/1.png | Bin 0 -> 19590 bytes media/1061/2.png | Bin 0 -> 33127 bytes media/1061/3.png | Bin 0 -> 27594 bytes 5 files changed, 121 insertions(+) create mode 100644 md/1061.md create mode 100644 media/1061/0.png create mode 100644 media/1061/1.png create mode 100644 media/1061/2.png create mode 100644 media/1061/3.png diff --git a/md/1061.md b/md/1061.md new file mode 100644 index 0000000..b7ea545 --- /dev/null +++ b/md/1061.md @@ -0,0 +1,121 @@ +ข้อนี้ให้ค่าอุณหภูมิในตาราง $N \times N$ $(N \leq 20)$ และกำหนดว่าจากช่องใดๆ จะสามารถขยับไปช่องรอบๆ ที่มีอุณหภูมิที่สูงกว่าเท่านั้น โดยมีบางช่องที่ค่าอุณหภูมิเป็น 100 ซึ่งจะเข้าไม่ได้เลย + +### แนวคิด + +สำหรับข้อนี้เราสามารถมองว่าแต่ละช่องของเป็นตารางที่ไม่ได้มีค่าอุณหภูมิเป็น 100 เป็น Node และระหว่างสองช่องที่ติดกันจะมี Edge ที่มีทิศทางจากช่องที่มีอุณหภูมิน้อยกว่าไปยังช่องที่มีอุณหภูมิมากกว่า + +จากนั้นหากทำ State Space Search เพื่อหาว่า Node ใดในตารางบ้างที่มี Path ที่เป็นไปได้จากช่องเริ่มต้นและเอาอันที่มีอุณหภูมิสูงสุดจะได้คำตอบที่ต้องการ + +เช่นในตัวอย่างประกอบในโจทย์สามารถวาด Edge ได้ดังในรูป + +![](../media/1061/0.png) + +### State Space Search + +State Space Search (การค้นหาในปริภูมิสถานะ) เป็นรูปแบบขั้นตอนวิธีที่จะแทน State (สถานะ) เป็น Node ใน Graph โดย Node $A$ จะมี Edge ไปยัง Node $B$ ก็ต่อเมื่อสามารถเปลี่ยนจาก State $A$ ไป $B$ ได้ในหนึ่งขั้น ใน State Space Search เราจะเริ่มจาก Node ใดๆ แล้วพิจารณา Node รอบข้างที่มี Edge เชื่อมเพื่อพิจารณาแต่ละ State ที่สามารถไปถึงจาก State เริ่ม สำหรับข้อนี้จะต้องหา State ที่มีค่าอุณหภูมิสูงสุด + +รูปประกอบต่อไปนี้เป็นตัวอย่างของ State Space + +![](../media/1061/1.png) + +ในตัวอย่างนี้จะเห็นได้ว่า State $B$ สามารถไป $D$ และ $D$ กับ $E$ สามารถไปมาซึ่งกันและกัน + +ในการทำ State Space Search จะมีวิธีพื้นฐานหลักๆ สองแบบคือ Depth First Search (DFS) กับ Breadth First Search (BFS) + +#### DFS + +ใน DFS จะทำการค้นหาในรูปแบบ Recursive โดยสำหรับแต่ละ State ที่พิจารณา จะ DFS ลงไปใน Node ที่มี Edge ที่ยังได้ไม่ถูกพิจารณา + +หากทำ DFS จาก State $A$ ในตัวอย่างก่อนหน้านี้จะได้ดังในรูป (ลูกศรสีเขียวแทนการเรียก DFS แบบ Recursive และลูกศรสีแดงแทนการ Return กลับมา) + +![](../media/1061/2.png) + +ในตัวอย่างนี้จะพิจารณา State ในลำดับ $A,B,D,E,C,F,G$ + +เมื่อนำมาใช้กับข้อนี้จะได้โค้ดดังนี้ + +```cpp +int M; +int T[22][22]; +int visited[22][22]; + +int dfs(int x, int y) { + visited[x][y] = 1; // visited[x][y] จะแทนว่าช่อง x,y ถูกพิจารณาแล้ว + + int ret = T[x][y]; + + int dx[4] = {0, 0, -1, 1}; + int dy[4] = {1, -1, 0, 0}; + for (int u = 0; u < 4; u++) { // พิจารณา 4 ทิศทาง + int ux = x + dx[u]; + int uy = y + dy[u]; + + if (1 <= ux && ux <= M && 1 <= uy && uy <= M && !visited[ux][uy] && + T[ux][uy] != 100 && + T[ux][uy] > T[x][y]) // ช่องใหม่ต้องอยู่ในกรอบ ยังไม่ถูกพิจารณา และ + // ต้องมีอุณหภูมิที่มากกว่าช่องปัจจุบันและไม่ใช่ 100 + ret = max(dfs(ux, uy), ret); // ค่าที่มากที่สุดจะเป็นค่าที่มากสุดของช่องนี้หรือช่องที่สามารถไปจากช่องนี้ + } + + return ret; +} +``` + +ในโค้ดตัวอย่างนี้จะพิจารณาแต่ละช่องที่อยู่ด้าน บน ล่าง ซ้าย หรือ ขวา ว่าเข้าเงื่อนไขที่จะสามารถไปจากช่องปัจจุบันไหมและหากสามารถไปจะเรียก DFS แบบ Recursive + +DFS จะใช้ Memory ใน Stack ตามความยาวของ Path ที่ทำ Recursion ซึ่งสำหรับข้อนี้เป็นได้อย่างมาก $\mathcal{O}(M^2)$ และจะใช้ Time Complexity ตามจำนวน Node และ Edge ที่ถูกพิจารณาซึ่งสำหรับข้อนี้เป็น $\mathcal{O}(M^2)$ เช่นกัน + +#### BFS + +สำหรับ BFS จะต่างจาก DFS ตรงที่เมื่อเจอ Edge ที่ไปได้แล้วจะไม่นำมาพิจารณาทันทีแต่ละนำเข้า Queue ก่อน โดยการพิจารณาแต่ละ Node จะเรียงตามลำดับใน Queue + +ตัวอย่างดังภาพ + +![](../media/1061/3.png) + +ในตัวอย่างนี้จะพิจารณาเป็นลำดับ $A,B,C,D,E,F,G$ ตามลูกศรสีเขียว + +โค้ด BFS ตัวอย่างสำหรับข้อนี้ + +```cpp +int M; +int T[22][22]; + +int bfs(int x, int y) { + int queued[22][22]; + queued[x][y] = 1; + std::deque> q; // Queue สำหรับการเก็บแต่ละ Node + + q.push_back(std::make_pair(x, y)); // นำ Node เริ่มต้นเข้า Queue + + int ret = T[x][y]; + + while (!q.empty()) { + int qx = q[0].first, qy = q[0].second; + q.pop_front(); // นำ Node แรกมาพิจารณาและเอาออกจาก Queue + + ret = std::max(ret, T[qx][qy]); + + int dx[4] = {0, 0, -1, 1}; + int dy[4] = {1, -1, 0, 0}; + for (int u = 0; u < 4; u++) { + int ux = qx + dx[u]; + int uy = qy + dy[u]; + + if (1 <= ux && ux <= M && 1 <= uy && uy <= M && !queued[ux][uy] && + T[ux][uy] != 100 && T[ux][uy] > T[qx][qy]) { + q.push_back(std::make_pair(ux, uy)); // ใส่ Node ใหม่ที่เจอเข้า Queue + queued[ux][uy] = 1; + } + } + } + + return ret; +} +``` + +สังเกตว่าในโค้ดนี้จะเรียก BFS เพียงรอบเดียวต่างจาก DFS โดยจะนำ Node มาใส่ Queue แทนการใช้ Recursion + +BFS จะใช้ Memory ตามจำนวน Node ที่อยู่ใน Queue และเวลาตามจำนวน Node / Edge ที่ถูกพิจารณา + 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