From b26a51459b449ba6f45b6f3b9b82752c36013367 Mon Sep 17 00:00:00 2001 From: "phluenam@gmail.com" Date: Mon, 30 Oct 2023 22:16:26 +0800 Subject: [PATCH 1/2] Add_1142 --- md/1142.md | 110 +++++++++++++++++++++++++++++++++++++++++++++++ media/1142/0.png | Bin 0 -> 85512 bytes media/1142/1.png | Bin 0 -> 92655 bytes media/1142/2.png | Bin 0 -> 100572 bytes 4 files changed, 110 insertions(+) create mode 100644 md/1142.md create mode 100644 media/1142/0.png create mode 100644 media/1142/1.png create mode 100644 media/1142/2.png diff --git a/md/1142.md b/md/1142.md new file mode 100644 index 0000000..dda3095 --- /dev/null +++ b/md/1142.md @@ -0,0 +1,110 @@ +ข้อนี้ให้บริษัทมา $N$ $(N\leq100000)$ บริษัทที่ต้องการจองศูนย์ประชุมซึ่งรองรับได้อย่างมากวันละ $K$ $(K\leq100)$ บริษัท + +บริษัท $i$ ต้องการจองศูนย์นี้ระหว่างวันที่ $X_i$ ถึง $Y_i$ $(1\leq X_i\leq Y_i\leq 1000000000)$ โดยการจองจะเรียงจาก $i=1$ ไปถึง $i=N$ หากวันใดมีบริษัทจองครบแล้ว $K$ บริษัท ศูนย์จะเต็มและไม่รับการจองของบริษัทต่อมาที่ขอจองในช่วงที่ทับวันดังกล่าว (กล่าวคือถ้าในช่วง $[X_i,Y_i]$ มีวันใดที่ครบ $K$ แล้วบริษัทที่ $i$ จะไม่ได้จองเลยทั้งช่วง $[X_i,Y_i]$) + +### แนวคิด + +อย่างแรกสังเกตว่าเราต้องสนใจเพียงวันที่ปรากฎใน $X_i$ หรือ $Y_i$ ใดๆ เป็นพิกัดเพราะหากมีการครบ $K$ วันจะต้องครบที่พิกัดดังกล่าว ดังนั้นสามารถทำ Coordinate Compression (เช่นเดียวกับใน https://programming.in.th/tasks/1138/solution) + +ในการพิจารณาแต่ละคำขอจองจากแต่ละบริษัทเราจะต้อง Query ว่าในช่วง $[X_i,Y_i]$ (ที่ทำ Coordinate Compression แล้ว) มีจองไปแล้วกี่บริษัทในวันที่ถูกจองมากที่สุด + +จากนั้นหากยังมีจองไม่ถึง $K$ จะต้อง Update ช่วงนี้ว่ามีจองเพิ่มขึ้น 1 บริษัทสำหรับทุกวันในช่วง $[X_i,Y_i]$ + +สังเกตว่า Operation ที่เราต้องการมีเพียง Query กับ Update คล้ายกับ Segment Tree เพียงแต่การ Update จะต้องรองรับการ Update ช่วงอย่างมีประสิทธิภาพด้วย โครงสร้างข้อมูลที่รองรับ Operation ดังกล่าวคือ Lazy Segment Tree + +### Lazy Segment Tree + +Lazy Segment Tree เป็นโครงสร้าง Segment Tree (อ่านเพิ่มได้จาก https://programming.in.th/tasks/1147/solution) ที่เพิ่มประสิทธิ์ภาพในการ Update แบบช่วง โดยในการ Update แบบช่วงจะเก็บเพิ่มอีกค่าในแต่ละ Node ซึ่งเป็นค่า "Lazy" ที่แทนว่าในทั้งช่วงนี้ยังค้าง Update อะไรอยู่ + +รูปต่อไปนี้เป็น Lazy Segment Tree ที่เริ่มด้วยค่า Lazy เป็น 0 ทั้งหมด + +![](../media/1142/0.png) + +#### Update + +สำหรับการ Update เพิ่มช่วง $[X_i,Y_i]$ ด้วยค่า $Z$ จะคล้ายๆ กับ Segment Tree ปกติ โดยต่างกันเพียงแค่ +- ถ้าช่วง $[l,r]$ ที่ Node นี้รับผิดชอบอยู่ใน $[X_i, Y_i]$ จะสามารถเพิ่มค่า Lazy ของ Node นี้ได้แล้ว return เลยโดยไม่ต้องลงไปใน Node ซ้ายหรือยวา +- ถ้า $[l,r]$ ตัดกับ $[X_i, Y_i]$ แต่ไม่ได้อยู่ข้างในทั้งหมดจะต้อง Push ค่า Lazy ไปยัง Node ซ้ายและขวาโดยใช้การ Update ด้วยค่า Lazy ดังกล่าว + +ตัวอย่างเช่น + +![](../media/1142/1.png) + +Node สีฟ้าคือ Node ที่ช่วงทั้งช่วง $[l,r]$ อยู่ใน Update ซึ่งจะโดนแก้เพียงค่า Lazy เช่นสำหรับ Node ที่คุมช่วง $C[1..3]$ สังเกตว่าสำหรับ Node เหล่านี้ Node ด้านล่างไม่ถูก Visit เลย + +ตัวอย่างโค้ด + +```cpp +int update(int X, int Y, int Z, int n, int l, int r) { + if (X <= l && r <= Y) { // [l,r] is contained in [X,Y] + Lazy[n] += Z; + return ST[n] + Lazy[n]; + } + if (r < X || Y < l) // [X,Y] is not in the range + return ST[n] + Lazy[n]; + + // [l,r] intersects [X,Y] + push_lazy(n, l, r); + + int mid = (l + r) / 2; + int new_left_value = update(X, Y, Z, n * 2, l, mid); + int new_right_value = update(X, Y, Z, n * 2 + 1, mid + 1, r); + + ST[n] = max(new_left_value, new_right_value); + return ST[n]; +} +``` + +#### Push Lazy +สำหรับการ Push ค่า Lazy เราเพียงต้อง Update Node ลูกซ้ายและขวาด้วยค่า Lazy ปัจจุบัน และแก้ค่า Lazy ให้เป็น $0$ เพื่อแสดงว่าไม่เหลือ Lazy Update ที่ค้างอยู่แล้ว เสร็จแล้วต้อง Update ค่า $\max$ ของช่วงที่เก็บไว้หลัง Update ลูก + +ตัวอย่างโค้ด + +```cpp +void push_lazy(int n, int l, int r) { + if (Lazy[n] == 0) + return; + + int mid = (l + r) / 2; + int new_left_value = update(l, r, Lazy[n], n * 2, l, mid); + int new_right_value = update(l, r, Lazy[n], n * 2 + 1, mid + 1, r); + Lazy[n] = 0; + ST[n] = max(new_left_value, new_right_value); +} +``` + +สังเกตว่าในการ Update แต่ละครั้งทุกขั้นตอนรวมถึงการ push_lazy จะใช้เวลา $\mathcal{O}(1)$ ดังนั้น Time Complexity จะเป็นไปตามจำนวน Node ที่ถูก Visit ซึ่งสามารถพิสูจน์ได้จำนวน Node ที่ต้อง Visit จะเป็น $\mathcal{O}(\log N)$ เช่นเดียวกับที่พิสูจน์ไว้แล้วสำหรับ Update ของ Segment Tree ทั่วไปใน https://programming.in.th/tasks/1147/solution + +#### Query + +สำหรับการ Query จะไม่ต่างกับ Query ของ Segment Tree ปกติมาก โดยเพียวต้องบวกค่า Lazy เข้าไปในค่าที่ Return หาก $[l,r]$ อยู่ในช่วง Query และต้อง push_lazy เช่นเดียวกับการ Update หาก $[l,r]$ ตัดกันช่วง Query แต่ไม่ได้อยู่ข้างในช่วงทั้งหมด + +ตัวอย่างประกอบ + +![](../media/1142/2.png) + +ในตัวอย่างนี้ค่าที่ถูก Return คือค่าที่มากสุดของ Node ที่อยู่ในช่วง (Node สีฟ้า) สังเกตว่า Node สำหรับ $C[4]$ กับ $C[5]$ ถูก push_lazy จากค่า Lazy ของ Node $C[4..5]$ + +ตัวอย่างโค้ด +```cpp +int query(int A, int B, int n, int l, int r) { + if (A <= l && r <= B) // [l,r] is a subset of [a,b] + return ST[n] + Lazy[n]; + if (B < l || r < A) // [l,r] does not intersect [a,b] + return -1000000001; // -inf + + // [l,r] intersects [a,b] + push_lazy(n, l, r); + + int mid = (l + r) / 2; + int left_query = query(A, B, n * 2, l, mid); + int right_query = query(A, B, n * 2 + 1, mid + 1, r); + + return max(left_query, right_query); +} +``` +การ Query จะใช้เวลา $\mathcal{O}(\log N)$ เช่นเดียวกับการ Update + +### Time Complexity + +ตามที่อธิบายไว้ด้านบนเราจะทำ Coordinate Compression ที่ใช้เวลา $\mathcal{O}(N\log N)$ แล้วใช้ Lazy Segment Tree สำหรับการ Update และ Query โดยแต่ละ Operation ใช้เวลา $\mathcal{O}(\log N)$ (เพราะจำนวนพิกัดที่ใช้ใน Lazy Segment Tree หลัง Compression คืออย่างมาก $2N$) ซึ่งต้องทำ $N$ ครั้ง จึงใช้เวลาทั้งหมด $\mathcal{O}(N\log N)$ diff --git a/media/1142/0.png b/media/1142/0.png new file mode 100644 index 0000000000000000000000000000000000000000..f00074744ff2cb7714a36bcc26a304f32e923b86 GIT binary patch literal 85512 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z0@#U^%*wuM{-+cF0!jj~coe7T{t$ovWkl|pLg)iR1e>D&bmDI&z#2&Li*FMD4}4&+ y4xDFg=E|t$e>(B=cDSAYPAm-d|A`a3?1}W0cDwS}?#40rCnKpKk#+y6_x}ZwnNVZ^ literal 0 HcmV?d00001 From 9f919ab1381c7d91704b034d606cb497de9040dd Mon Sep 17 00:00:00 2001 From: "phluenam@gmail.com" Date: Mon, 30 Oct 2023 22:29:29 +0800 Subject: [PATCH 2/2] edit with ` char --- md/1142.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/md/1142.md b/md/1142.md index dda3095..5cdfb84 100644 --- a/md/1142.md +++ b/md/1142.md @@ -22,9 +22,9 @@ Lazy Segment Tree เป็นโครงสร้าง Segment Tree (อ่ #### Update -สำหรับการ Update เพิ่มช่วง $[X_i,Y_i]$ ด้วยค่า $Z$ จะคล้ายๆ กับ Segment Tree ปกติ โดยต่างกันเพียงแค่ +สำหรับการ `update` เพิ่มช่วง $[X_i,Y_i]$ ด้วยค่า $Z$ จะคล้ายๆ กับ Segment Tree ปกติ โดยต่างกันเพียงแค่ - ถ้าช่วง $[l,r]$ ที่ Node นี้รับผิดชอบอยู่ใน $[X_i, Y_i]$ จะสามารถเพิ่มค่า Lazy ของ Node นี้ได้แล้ว return เลยโดยไม่ต้องลงไปใน Node ซ้ายหรือยวา -- ถ้า $[l,r]$ ตัดกับ $[X_i, Y_i]$ แต่ไม่ได้อยู่ข้างในทั้งหมดจะต้อง Push ค่า Lazy ไปยัง Node ซ้ายและขวาโดยใช้การ Update ด้วยค่า Lazy ดังกล่าว +- ถ้า $[l,r]$ ตัดกับ $[X_i, Y_i]$ แต่ไม่ได้อยู่ข้างในทั้งหมดจะต้อง Push ค่า Lazy ไปยัง Node ซ้ายและขวาโดยใช้การ `update` ด้วยค่า Lazy ดังกล่าว ตัวอย่างเช่น @@ -56,7 +56,7 @@ int update(int X, int Y, int Z, int n, int l, int r) { ``` #### Push Lazy -สำหรับการ Push ค่า Lazy เราเพียงต้อง Update Node ลูกซ้ายและขวาด้วยค่า Lazy ปัจจุบัน และแก้ค่า Lazy ให้เป็น $0$ เพื่อแสดงว่าไม่เหลือ Lazy Update ที่ค้างอยู่แล้ว เสร็จแล้วต้อง Update ค่า $\max$ ของช่วงที่เก็บไว้หลัง Update ลูก +สำหรับการ Push ค่า Lazy เราเพียงต้อง `update` Node ลูกซ้ายและขวาด้วยค่า Lazy ปัจจุบัน และแก้ค่า Lazy ให้เป็น $0$ เพื่อแสดงว่าไม่เหลือ Lazy Update ที่ค้างอยู่แล้ว เสร็จแล้วต้องแก้ค่า $\max$ ของช่วงที่เก็บไว้เพื่อให้ถูกต้องหลัง `update` ลูก ตัวอย่างโค้ด @@ -73,17 +73,17 @@ void push_lazy(int n, int l, int r) { } ``` -สังเกตว่าในการ Update แต่ละครั้งทุกขั้นตอนรวมถึงการ push_lazy จะใช้เวลา $\mathcal{O}(1)$ ดังนั้น Time Complexity จะเป็นไปตามจำนวน Node ที่ถูก Visit ซึ่งสามารถพิสูจน์ได้จำนวน Node ที่ต้อง Visit จะเป็น $\mathcal{O}(\log N)$ เช่นเดียวกับที่พิสูจน์ไว้แล้วสำหรับ Update ของ Segment Tree ทั่วไปใน https://programming.in.th/tasks/1147/solution +สังเกตว่าในการ `update` แต่ละครั้งทุกขั้นตอนรวมถึงการ `push_lazy` จะใช้เวลา $\mathcal{O}(1)$ ดังนั้น Time Complexity จะเป็นไปตามจำนวน Node ที่ถูก Visit ซึ่งสามารถพิสูจน์ได้จำนวน Node ที่ต้อง Visit จะเป็น $\mathcal{O}(\log N)$ เช่นเดียวกับที่พิสูจน์ไว้แล้วสำหรับ `update` ของ Segment Tree ทั่วไปใน https://programming.in.th/tasks/1147/solution #### Query -สำหรับการ Query จะไม่ต่างกับ Query ของ Segment Tree ปกติมาก โดยเพียวต้องบวกค่า Lazy เข้าไปในค่าที่ Return หาก $[l,r]$ อยู่ในช่วง Query และต้อง push_lazy เช่นเดียวกับการ Update หาก $[l,r]$ ตัดกันช่วง Query แต่ไม่ได้อยู่ข้างในช่วงทั้งหมด +สำหรับการ `query` จะคล้าย `query` ของ Segment Tree ปกติ โดยเพียงต้องบวกค่า Lazy เข้าไปในค่าที่ Return หาก $[l,r]$ อยู่ในช่วง Query และต้อง `push_lazy` เช่นเดียวกับการ `update` หาก $[l,r]$ ตัดกันช่วง Query แต่ไม่ได้อยู่ข้างในช่วงทั้งหมด ตัวอย่างประกอบ ![](../media/1142/2.png) -ในตัวอย่างนี้ค่าที่ถูก Return คือค่าที่มากสุดของ Node ที่อยู่ในช่วง (Node สีฟ้า) สังเกตว่า Node สำหรับ $C[4]$ กับ $C[5]$ ถูก push_lazy จากค่า Lazy ของ Node $C[4..5]$ +ในตัวอย่างนี้ค่าที่ถูก Return คือค่าที่มากสุดของ Node ที่อยู่ในช่วง (Node สีฟ้า) สังเกตว่า Node สำหรับ $C[4]$ กับ $C[5]$ ถูก `push_lazy` จากค่า Lazy ของ Node $C[4..5]$ ตัวอย่างโค้ด ```cpp @@ -103,8 +103,8 @@ int query(int A, int B, int n, int l, int r) { return max(left_query, right_query); } ``` -การ Query จะใช้เวลา $\mathcal{O}(\log N)$ เช่นเดียวกับการ Update +การ `query` จะใช้เวลา $\mathcal{O}(\log N)$ เช่นเดียวกับการ `update` ### Time Complexity -ตามที่อธิบายไว้ด้านบนเราจะทำ Coordinate Compression ที่ใช้เวลา $\mathcal{O}(N\log N)$ แล้วใช้ Lazy Segment Tree สำหรับการ Update และ Query โดยแต่ละ Operation ใช้เวลา $\mathcal{O}(\log N)$ (เพราะจำนวนพิกัดที่ใช้ใน Lazy Segment Tree หลัง Compression คืออย่างมาก $2N$) ซึ่งต้องทำ $N$ ครั้ง จึงใช้เวลาทั้งหมด $\mathcal{O}(N\log N)$ +ตามที่อธิบายไว้ด้านบนเราจะทำ Coordinate Compression ที่ใช้เวลา $\mathcal{O}(N\log N)$ แล้วใช้ Lazy Segment Tree สำหรับการ `update` และ `query` โดยแต่ละ Operation ใช้เวลา $\mathcal{O}(\log N)$ (เพราะจำนวนพิกัดที่ใช้ใน Lazy Segment Tree หลัง Compression คืออย่างมาก $2N$) ซึ่งต้องทำ $N$ ครั้ง จึงใช้เวลาทั้งหมด $\mathcal{O}(N\log N)$