Topological sort and PERT

Author: zkeulr

ECE27000/ECE27000.pdf

@@ -0,0 +1,57 @@
+\documentclass[8pt]{article}
+\usepackage{amssymb}
+\usepackage{hyperref}
+\usepackage{pgfplots}
+\usepackage{placeins}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{circuitikz}
+\usetikzlibrary{circuits.logic.US, circuits.ee.IEC, positioning}
+\usepackage{amsmath}
+\usepackage{graphicx}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{listings}
+\usepackage{xcolor}
+\usepackage{geometry}
+\input{kvmacros.tex}
+\geometry{margin=0.4in}
+
+\lstdefinelanguage{SystemVerilog}{
+  morekeywords={module, endmodule, logic, bit, int, enum, struct,
+    always_ff, always_comb, initial, final, interface, modport, property,
+    assert, class, rand, constraint, generate, endgenerate, if, else, begin, end},
+  sensitive=true,
+  morecomment=[l]{//},
+  morecomment=[s]{/*}{*/},
+  morestring=[b]",
+}
+
+\lstset{
+  language=SystemVerilog,
+  basicstyle=\ttfamily\footnotesize,
+  keywordstyle=\color{blue}\bfseries,
+  commentstyle=\color{gray}\itshape,
+  stringstyle=\color{red},
+  frame=single,
+  breaklines=true,
+  postbreak=\mbox{\textcolor{red}{$\hookrightarrow$}\space},
+}
+
+\begin{document}
+
+\begin{itemize}
+    \item Setup time $t_{setup}$: time before clock edge that data must be
+          stable.
+    \item Hold time $t_{hold}$: time after clock edge that data must be stable.
+    \item Propagation delay $t_{pcq}$: time after clock edge that output
+          is guaranteed to be stable.
+    \item Contamination delay $t_{ccq}$: time after clock edge that output
+          might be unstable.
+\end{itemize}
+
+\begin{align}
+    T_{clk} \geq t_{pcq} + t_{pd} + t_{setup} & \text{Setup Time Constraint} \\
+    t_{hold} < t_{ccq} + t_{cd}               & \text{Hold Time Constraint}
+\end{align}
+\end{document}

ECE27000/midterm2.pdf

@@ -209,4 +209,37 @@ \subsection{Shortest Path Algorithms}
 \end{lstlisting}
 Bellman-Ford can be optimized to stop early once the optimal paths have
 been reached. If there is no change after two iterations in a row, then
-the best solution has been found and the algorithm can terminate early.
+the best solution has been found and the algorithm can terminate early.
+
+Given a directed acyclic graph (DAG), finding the shortest path is as
+easy as following these steps:
+\marginnote{A topological sort of an acyclic directed graph is an ordering of its
+    vertices such that for every directed edge $<u,v>$ from vertex $u$ to
+    vertex $v$, $u$ comes before $v$ in the ordering. Both depth-first and
+    breadth-first sorting works.}
+\begin{lstlisting}[label={DAG Shortest Path Algorithm}]
+    DAG-Shortest-Paths(G, w, s) // s: source
+    Topologically sort V[G]
+    for each node u in V[G]
+        d[u] <- INFTY; pred[u] <- NULL
+    d[s] <- 0
+    for each node u in topological order
+        for each v in adj[u]
+            if (d[v] > d[u] + w<u,v>)
+                d[v] <- d[u] + w<u,v>
+                pred[v] <- u
+\end{lstlisting}
+
+Finding the longest path is as easy as initializing distances to negative infinity
+and changing the $>$ to a $<$.
+\marginnote{In general, finding the longest path is an NP hard problem. DAGs are
+    a special case.}
+
+
+\subsection{PERT Diagrams}
+Project Evaluation and Review Technique (PERT) diagrams are represented as graphs.
+Each project has a number of tasks, each with duration D.
+Each task is a node in the graph. An edge $<i, j>$ represents that the task $j$ is a
+successor of task $i$ and cannot begin until task $i$ is completed. A critical path
+is one where each task is critical. A critical task is one that if its completion is
+delayed, project completion is delayed.