From 1334745a580044daec494ae843eee88154bed0de Mon Sep 17 00:00:00 2001
From: Filipe Rodrigues <filipejacintorodrigues1@gmail.com>
Date: Sat, 6 Jan 2024 16:17:36 +0000
Subject: [PATCH] Mostly finished ex5.

---
 .vscode/settings.json  |  1 +
 code/5.R               |  4 +--
 typst/appendixes/5.typ |  6 +++++
 typst/exercises/5.typ  | 56 ++++++++++++++++++++++++++++++++++++++++++
 typst/main.typ         | 10 ++++++++
 5 files changed, 74 insertions(+), 3 deletions(-)
 create mode 100644 typst/appendixes/5.typ
 create mode 100644 typst/exercises/5.typ

diff --git a/.vscode/settings.json b/.vscode/settings.json
index c99f5e4..912a180 100644
--- a/.vscode/settings.json
+++ b/.vscode/settings.json
@@ -1,6 +1,7 @@
 {
   "cSpell.words": [
     "byrow",
+    "forall",
     "ggplot",
     "ggsave",
     "linebreak",
diff --git a/code/5.R b/code/5.R
index 0cbdf4d..a3dfdfb 100644
--- a/code/5.R
+++ b/code/5.R
@@ -2,14 +2,12 @@ source("code/util/network_4_5.R")
 source("code/util/shortest_path.R")
 require(lpSolve)
 
-numServers <- 1
+numServers <- 5
 numNodes <- nrow(L)
 
-
 Costs <- L
 Costs[L == 0] <- Inf
 
-# Delays are deltas
 Delays <- matrix(rep(0, len = numNodes^2), nrow = numNodes)
 for (i in 1:numNodes) {
   for (j in 1:numNodes) {
diff --git a/typst/appendixes/5.typ b/typst/appendixes/5.typ
new file mode 100644
index 0000000..7d8637b
--- /dev/null
+++ b/typst/appendixes/5.typ
@@ -0,0 +1,6 @@
+#import "/typst/util.typ" as util: indent_par, code_figure
+
+#code_figure(
+  text(size: 0.8em, raw(read("/code/5.R"), lang: "R", block: true)),
+  caption: [Code used for exercise 5],
+)
diff --git a/typst/exercises/5.typ b/typst/exercises/5.typ
new file mode 100644
index 0000000..4e7e9d6
--- /dev/null
+++ b/typst/exercises/5.typ
@@ -0,0 +1,56 @@
+#import "/typst/util.typ" as util: indent_par, code_figure
+
+
+#indent_par[In order to optimize the server locations, we've used the `lpSolve` package from `R`.]
+
+#indent_par[First, we declared the following variables:]
+
+- Number of servers, $z_i$, in node $i$.
+- Replica, $g_i^j$, has node $i$ be served by node $j$
+
+
+#indent_par[Afterwards, we can construct the following equations:]
+
+#figure(
+	pad(1em, table(
+		columns: (auto, 1fr),
+		align: left + horizon,
+
+		[Description], [Equation],
+		[Have at most `numServers` replicas], [$sum_i z_i = "numServers"$],
+		[For each node, it must be served by exactly 1 replica], [$forall_i sum_j g_i^j == 1$],
+		[If a node isn't a replica, no other node can be served by it.], [$forall_(i,j) g_i^j <= z_j$],
+	)),
+	kind: table,
+	caption: [Problem equations]
+)
+
+#indent_par[Our objective function is the following in equation 2:]
+
+$ sum_i,j g_i^j dot C_i^j $
+
+#indent_par[Where $C_i^j$ is the delay between nodes $i$ and $j$.]
+
+#indent_par[The code used to implement this is present in the appendix.]
+
+#pagebreak()
+
+#indent_par[After running our code for several different number of servers, we obtained the following results in table 14:]
+
+#figure(
+	pad(1em, table(
+		columns: (auto, auto, auto),
+		align: left + horizon,
+
+		[Servers], [Replica servers], [Cost],
+		[1], [8], [3862.00],
+		[2], [5, 15], [2412.00],
+		[3], [5, 12, 13], [1832.00],
+		[4], [3, 5, 10, 13], [1521.00],
+		[5], [3, 5, 11, 12, 15], [1269.00],
+	)),
+	kind: table,
+	caption: [Problem equations]
+)
+
+#indent_par[From this we can conclude that more replicas leads to a lower overall cost.]
diff --git a/typst/main.typ b/typst/main.typ
index 8236599..b803201 100644
--- a/typst/main.typ
+++ b/typst/main.typ
@@ -56,9 +56,16 @@
 === 3. Exercise 3
 #include "exercises/3.typ"
 
+== B. Heuristic methods
+
 === 4. Exercise 4
 #include "exercises/4.typ"
 
+== C. Replica placement
+
+=== 5. Exercise 5
+#include "exercises/5.typ"
+
 #pagebreak()
 
 = Appendix
@@ -74,3 +81,6 @@
 
 === 4. Exercise 4
 #include "appendixes/4.typ"
+
+=== 5. Exercise 5
+#include "appendixes/5.typ"