From 2870d40f6b888cb1d91c1d8ed33fbb8287a3aa04 Mon Sep 17 00:00:00 2001 From: Renge Date: Sat, 3 Sep 2022 21:05:06 -0400 Subject: [PATCH] add: hw1 --- HW1/2_1_1.png | Bin 0 -> 177674 bytes HW1/2_1_2.png | Bin 0 -> 192630 bytes HW1/2_1_3.png | Bin 0 -> 190469 bytes HW1/2_2_1.png | Bin 0 -> 224605 bytes HW1/2_2_2.png | Bin 0 -> 244247 bytes HW1/HW1.pdf | Bin 0 -> 1402658 bytes HW1/hw1-polygon1-guard.ipe | 464 ++++++++++++++++++++ HW1/hw1-polygon2-guard.ipe | 521 ++++++++++++++++++++++ HW1/hw1.html | 853 +++++++++++++++++++++++++++++++++++++ HW1/hw1.md | 38 ++ 10 files changed, 1876 insertions(+) create mode 100644 HW1/2_1_1.png create mode 100644 HW1/2_1_2.png create mode 100644 HW1/2_1_3.png create mode 100644 HW1/2_2_1.png create mode 100644 HW1/2_2_2.png create mode 100644 HW1/HW1.pdf create mode 100644 HW1/hw1-polygon1-guard.ipe create mode 100644 HW1/hw1-polygon2-guard.ipe create mode 100644 HW1/hw1.html create mode 100644 HW1/hw1.md diff --git a/HW1/2_1_1.png b/HW1/2_1_1.png new 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HW1

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Problem 1

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      In this graph, there are 3 forced diagonals (7-12, 6-14 and 5-17) which divide this graph in to 4 parts. For the right most part it is a blunt nose fox so it has 4 ways of triangulations; for the second one there is only one way because it is a fox; the third one is again, a blunt nose fox with 4 ways; as for the last part we have to do case analysis:

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        1. use 4-17 but not 2-5: there are two convex polygon with 3 and 6 vertices, so there are 14*1=14 ways to triangulation
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        1. use 2-5 but not 4-17: for the graph under 2-5, it is a convex polygon with 4 vertices so there are 2 ways; for the upper part, 2-17 is a forced diagonal so it can be treated as a 4 vertices polygon so overall it has 2*2=4 ways
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        1. not use both 4-17 and 2-5: for the last situation, 3-17 is a forced diagonal so it has 5*1*1=5 ways
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      Overall, there are 4*1*4*(14+4+5)=368 distinct ways to triangulation this graph

      +
    2. +
    +
  • +
  • +
      +
    1. +

      In this graph, there are a pair of diagonals that are mutual exclusive: 1-6 and 0-2, if we do case analysis based on that:

      +
        +
      • +
          +
        1. use 1-6: the left side is a convex polygon with 8 vertices so it has 6 ways; the right side is a blunt nose fox so it has 4 ways
        2. +
        +
      • +
      • +
          +
        1. use 0-2: then there are two forced diagonals (0-6 and 2-6). If we connect them, then there is a convex polygon with 7 vertices on the left and a fox on the right. In total, there should be 1*1*1*42 ways
        2. +
        +
      • +
      +

      Over all, there are 132*4+42=570 distinct ways of triangulations.

      +
    2. +
    +
  • +
+

Problem 2

+
    +
  • +
      +
    1. +
        +
      • +
          +
        1. 2_1_1
        2. +
        +
      • +
      • +
          +
        1. As the graph above, there are 6 reds, 6 blues and 7 yellows, then we put guards on either reds or blues and we can get 6 guards.
        2. +
        +
      • +
      • +
          +
        1. +

          2_1_2 +If we put witness points near 18, 2, 7 and 10, and in between 4 and 5, we got 5 areas that does not overlap on any vertex, w(P)5w(P) \geq 5. Also, we can find 5 guards A, B, C, D and E that cover P, thus g(P)5g(P) \leq 5

          +

          5w(p)g(p)55 \leq w(p) \leq g(p) \leq 5, and we can get g(p)=5g(p) = 5

          +
        2. +
        +
      • +
      • +
          +
        1. +

          2_1_3 +If we put witness points near 18, 2 and 7, and in between 4 and 5, we got 4 areas that does not overlap, so w(P)4w(P) \geq 4. Also, we can find 4 guards A, B, C and D that cover P, thus g(P)4g(P) \leq 4

          +

          4w(p)g(p)44 \leq w(p) \leq g(p) \leq 4, and we can get g(p)=4g(p) = 4

          +
        2. +
        +
      • +
      +
    2. +
    +
  • +
  • +
      +
    1. +
        +
      • +
          +
        1. 2_2_1
        2. +
        +
      • +
      • +
          +
        1. As the graph above, there are 8 reds, 8 blues and 8 yellows, then we put guards on any of reds, yellows or blues and we can get 8 guards.
        2. +
        +
      • +
      • +
          +
        1. +

          2_2_2 +If we put witness points near 3, 10, 17 and 23, and in between 13 and 14, we got 5 areas that does not overlap on any vertex, w(P)5w(P) \geq 5. Also, we can find 5 guards A, B, C, D and E that cover P, thus g(P)5g(P) \leq 5

          +

          5w(p)g(p)55 \leq w(p) \leq g(p) \leq 5, and we can get g(p)=5g(p) = 5

          +
        2. +
        +
      • +
      • +
          +
        1. +

          2_2_2 +If we put witness points near 3, 10, 17 and 23, and in between 13 and 14, we got 5 areas that does not overlap, w(P)5w(P) \geq 5. Also, we can find 5 guards A, B, C, D and E that cover P, thus g(P)5g(P) \leq 5

          +

          5w(p)g(p)55 \leq w(p) \leq g(p) \leq 5, and we can get g(p)=5g(p) = 5

          +
        2. +
        +
      • +
      +
    2. +
    +
  • +
+ + + + + \ No newline at end of file diff --git a/HW1/hw1.md b/HW1/hw1.md new file mode 100644 index 0000000..9166fb2 --- /dev/null +++ b/HW1/hw1.md @@ -0,0 +1,38 @@ +# HW1 +## Problem 1 + - 1. In this graph, there are 3 forced diagonals (7-12, 6-14 and 5-17) which divide this graph in to 4 parts. For the right most part it is a blunt nose fox so it has 4 ways of triangulations; for the second one there is only one way because it is a fox; the third one is again, a blunt nose fox with 4 ways; as for the last part we have to do case analysis: + - 1. use 4-17 but not 2-5: there are two convex polygon with 3 and 6 vertices, so there are 14*1=14 ways to triangulation + - 2. use 2-5 but not 4-17: for the graph under 2-5, it is a convex polygon with 4 vertices so there are 2 ways; for the upper part, 2-17 is a forced diagonal so it can be treated as a 4 vertices polygon so overall it has 2*2=4 ways + - 3. not use both 4-17 and 2-5: for the last situation, 3-17 is a forced diagonal so it has 5\*1\*1=5 ways + + Overall, there are 4\*1\*4\*(14+4+5)=368 distinct ways to triangulation this graph + + - 2. In this graph, there are a pair of diagonals that are mutual exclusive: 1-6 and 0-2, if we do case analysis based on that: + - 1. use 1-6: the left side is a convex polygon with 8 vertices so it has 6 ways; the right side is a blunt nose fox so it has 4 ways + - 2. use 0-2: then there are two forced diagonals (0-6 and 2-6). If we connect them, then there is a convex polygon with 7 vertices on the left and a fox on the right. In total, there should be 1\*1\*1\*42 ways + + Over all, there are 132*4+42=570 distinct ways of triangulations. + +## Problem 2 + - 1. + - 1. ![2_1_1](2_1_1.png) + - 2. As the graph above, there are 6 reds, 6 blues and 7 yellows, then we put guards on either reds or blues and we can get 6 guards. + - 3. ![2_1_2](2_1_2.png) + If we put witness points near 18, 2, 7 and 10, and in between 4 and 5, we got 5 areas that does not overlap on any vertex, $w(P) \geq 5$. Also, we can find 5 guards A, B, C, D and E that cover P, thus $g(P) \leq 5$ + + $5 \leq w(p) \leq g(p) \leq 5$, and we can get $g(p) = 5$ + - 4. ![2_1_3](2_1_3.png) + If we put witness points near 18, 2 and 7, and in between 4 and 5, we got 4 areas that does not overlap, so $w(P) \geq 4$. Also, we can find 4 guards A, B, C and D that cover P, thus $g(P) \leq 4$ + + $4 \leq w(p) \leq g(p) \leq 4$, and we can get $g(p) = 4$ + - 2. + - 1. ![2_2_1](2_2_1.png) + - 2. As the graph above, there are 8 reds, 8 blues and 8 yellows, then we put guards on any of reds, yellows or blues and we can get 8 guards. + - 3. ![2_2_2](2_2_2.png) + If we put witness points near 3, 10, 17 and 23, and in between 13 and 14, we got 5 areas that does not overlap on any vertex, $w(P) \geq 5$. Also, we can find 5 guards A, B, C, D and E that cover P, thus $g(P) \leq 5$ + + $5 \leq w(p) \leq g(p) \leq 5$, and we can get $g(p) = 5$ + - 4. ![2_2_2](2_2_2.png) + If we put witness points near 3, 10, 17 and 23, and in between 13 and 14, we got 5 areas that does not overlap, $w(P) \geq 5$. Also, we can find 5 guards A, B, C, D and E that cover P, thus $g(P) \leq 5$ + + $5 \leq w(p) \leq g(p) \leq 5$, and we can get $g(p) = 5$ \ No newline at end of file

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