algo: add hw9

fix heading issue

7e1810-algo_hw/hw7.typ

@@ -9,23 +9,23 @@ Due: 2024.04.28
   ]
 }
 
-== Question 21.1-1
+=== Question 21.1-1
 
 Let $(u,v)$ be a minimum-weight edge in a connected graph $G$. Show that $(u,v)$ belongs to some minimum spanning tree of $G$.
 
 #ans[
   Let $T$ be a minimum spanning tree of $G$. If $(u,v)$ is not in $T$, then $T union \{(u,v)\}$ contains a cycle $C$. Since $(u,v)$ is the minimum-weight edge in $G$ that crosses the cut $(V(T), V - V(T))$, we can replace an edge in $C$ with $(u,v)$ to get a spanning tree $T'$ with $w(T') < w(T)$, which contradicts the assumption that $T$ is a minimum spanning tree.
 ]
 
-== Question 21.2-1
+=== Question 21.2-1
 
 Kruskal's algorithm can return different spanning trees for the same input graph $G$, depending on how it breaks ties when the edges are sorted into order. *Show that for each minimum spanning tree $T$ of $G$, there is a way to sort the edges of $G$ in Kruskal's algorithm so that the algorithm returns $T$.*
 
 #ans[
   Let $T$ be a minimum spanning tree of $G$. We sort the edges of $G$ in nondecreasing order of their weights. If there are ties, we break them arbitrarily. Since $T$ is a minimum spanning tree, the edges of $T$ are sorted before the edges not in $T$. Therefore, Kruskal's algorithm will add the edges of $T$ to the tree before adding any other edges, and the result will be $T$.
 ]
 
-== Question 21.2-4
+=== Question 21.2-4
 
 Suppose that all edge weights in a grpah are integers in the range from $1$ to $abs(V)$. How fast can you make Kruskal's algorithm run? What if the edge weights are integers in the range from 1 to $W$ for some constant $W$?
 

7e1810-algo_hw/hw8.typ

@@ -9,7 +9,7 @@ Due: 2024.05.05
   ]
 }
 
-== Exerciese 1
+=== Exerciese 1
 Proof that Bellman-Ford maximizes $x_1+x_2+dots.c+x_n$ subject to the constraints $x_j - x_i <= w_(i j)$ for all edges $(i,j)$ and $x <= 0$, and also minmizes $max_i {x_i}-min_i {x_i}$.
 
 #ans[
@@ -28,23 +28,23 @@ Proof that Bellman-Ford maximizes $x_1+x_2+dots.c+x_n$ subject to the constraint
   To see why this also minmizes $max_i {x_i}-min_i {x_i}$, we can see that the longest path is the one that maximizes the difference between the maximum and minimum values. (Suppose there exists another set of solutions that has a larger difference, say $max_i {x_i '} - min_i {x_i '}$, the constraint between the two sets of solutions would be violated, since the longest path is the one that maximizes the difference.)
 ]
 
-== Question 23.2-6
+=== Question 23.2-6
 
 Show how to use the output of the Floyd-Warshall algorithm to detect presence of a negative-weight cycle.
 
 #ans[
   If the output of the Floyd-Warshall algorithm contains a negative number on the diagonal, then there exists a negative-weight cycle in the graph. This is because the diagonal represents the shortest path from a vertex to itself, and if there exists a negative-weight cycle, then the shortest path from a vertex to itself would be negative infinity.
 ]
 
-== Question 23.3-4
+=== Question 23.3-4
 
 Professor Greenstreet claims that there is a simpler way to reweight edges than the method used in Johnson's algorithm. Letting $w^*=min_((u,v)in E){w(u,v)}$, just define $hat(w) (u,v)=w(u,v)-w^*$ for all edges $(u,v)in E$. What is wrong with the professor's method of reweighting?
 
 #ans[
   Might result in path with more edges longer than direct edge, which is not optimal.
 ]
 
-== Question 24.3-3
+=== Question 24.3-3
 
 Let $G=(V,E)$ be a bipartite graph with vertex partition $V=L union R$, and let $G$ be its corresponding flow network. Give a good upper bound on the length of any augmenting path found in $G$ during the execution of FORD-FULKERSON.
 

7e1810-algo_hw/hw9.typ

@@ -0,0 +1,35 @@
+== HW9 (Week 11)
+Due: 2024.05.19
+
+#let ans(it) = {
+  box(inset: 1em, width: 100%)[
+    #text(fill: blue)[
+      #it
+    ]
+  ]
+}
+
+=== Question 32.4-1
+Compute the prefix function $pi$ for the pattern `ababbabbabbababbabb`.
+
+#ans[
+  $
+    pi={0,0,1,2,0,1,2,0,1,2,0,1,2,3,4,5,6,7,8}
+  $
+]
+
+=== Question 32.4-6
+Show how to improve KMP-MATCHER by replacing the occurrence of $pi$ in line 5(but now line 10) by $pi'$, where $pi'$ is defined recusively for $q=1,2...,m-1$ by the equation
+$
+  pi'[q]=cases(0 quad & "if" pi[q]=0, pi'[pi[q]] quad &"if" pi[q]!=0 "and" P[pi[q]+1]=P[q+1], pi[q] quad & "otherwise")
+$
+
+Explain why the modified algorithm is correct, and explain in what sense this change constitutes an improvement.
+
+#ans[
+  If $P[q+1]!=T[i] "and" P[pi[q]+q]=P[q+1]!=T[i]$, there's no need to compare $P[pi[q]+q]$ with $T[i]$, because $P[pi[q]+q]$ is the same as $P[q+1]$, so we can directly compare $P[q+1]$ with $T[i]$. This change improves the efficiency of the algorithm.
+]
+
+#align(center)[
+  #image("imgs/sticker_2.jpg", width: 50%)
+]

7e1810-algo_hw/main.typ

@@ -7,6 +7,8 @@
 
 PB21000030 马天开
 
+#include "hw9.typ"
+#pagebreak()
 #include "hw8.typ"
 #pagebreak()
 #include "hw7.typ"