Graph alg explanations

Author: lee-naish

src/algorithms/explanations/ASTExp.md

@@ -32,8 +32,8 @@ As these algorithms execute, we can classify nodes into three sets.
 These are:
 
 
-- "Finalised" nodes, for which the shortest or least costly path back to the start node has already
-been finalised, that is the final parent node has been determined and is recorded;
+- "Finalised" nodes, for which the path back to the start node has
+been finalised, that is, the final parent node has been determined and recorded;
 
 - "Frontier" nodes, that are not finalised but are connected to a finalised node by a single edge; and
 
@@ -46,8 +46,8 @@ The frontier initially contains just the start node. The algorithms repeatedly
 pick a frontier node, finalise the node (its current parent becomes
 its final parent) and update information about the neighbours of the node.
 A\* uses a priority queue for the frontier nodes,
-ordered on the shortest distance to the node found so far *plus* the
-heuristic value of the node.  At each
+ordered on the shortest distance from the start node
+to the node found so far *plus* the heuristic value of the node.  At each
 stage the node with the minimum cost
 is removed for processing, and its neighbors have their information
 updated if a shorter path has now been found.

src/algorithms/explanations/BFSExp.md

@@ -29,8 +29,9 @@ As these algorithms execute, we can classify nodes into three sets.
 These are:
 
 
-- "Finalised" nodes, for which the shortest or least costly path back to the start node has already
-been finalised, that is the final parent node has been determined and is recorded;
+- "Finalised" nodes, for which the path back to the start node has 
+been finalised, that is, the final parent node has been determined and recorded;
+
 
 - "Frontier" nodes, that are not finalised but are connected to a finalised node by a single edge; and
 

src/algorithms/explanations/DFSExp.md

@@ -30,10 +30,9 @@ As these algorithms execute, we can classify nodes into three sets.
 These are:
 
 
-- "Finalised" nodes, for which the shortest or least costly path back to the start node has already
-been finalised, that is the final parent node has been determined and is
-recorded (DFS is an exception in that path lengths/costs are ignored and
-finalised nodes can have very long paths to them);
+- "Finalised" nodes, for which the path back to the start node has 
+been finalised, that is, the final parent node has been determined and recorded;
+
 
 - "Frontier" nodes, that are not finalised but are connected to a finalised node by a single edge; and
 

src/algorithms/explanations/DFSrecExp.md

@@ -14,7 +14,7 @@ This recursive coding is very simple: the core is a recursive
 function that is called with a node n and a parent p. If the parent of
 n has not yet been decided it assigns p as the parent and recursively
 calls the function on all nodes neighbouring n, with n as the parent,
-until an end node is found or all nodes in the componenent have been
+until an end node is found or all nodes in the component have been
 examined.
 
 Elsewhere we have an iterative version of DFS.
@@ -29,10 +29,10 @@ As these algorithms execute, we can classify nodes into three sets.
 These are:
 
 
-- "Finalised" nodes, for which the shortest or least costly path back to the start node has already
-been finalised, that is the final parent node has been determined and is
-recorded (DFS is an exception in that path lengths/costs are ignored and
-finalised nodes can have very long paths to them);
+- "Finalised" nodes, for which the path back to the start node has 
+been finalised, that is, the final parent node has been determined and
+recorded;
+
 
 - "Frontier" nodes, that are not finalised but are connected to a finalised node by a single edge; and
 

src/algorithms/explanations/DIJKExp.md

@@ -29,8 +29,9 @@ As these algorithms execute, we can classify nodes into three sets.
 These are:
 
 
-- "Finalised" nodes, for which the shortest or least costly path back to the start node has already
-been finalised, that is the final parent node has been determined and is recorded;
+- "Finalised" nodes, for which the path back to the start node has 
+been finalised, that is, the final parent node has been determined and recorded;
+
 
 - "Frontier" nodes, that are not finalised but are connected to a finalised node by a single edge; and
 
@@ -43,7 +44,8 @@ The frontier initially contains just the start node. The algorithms repeatedly
 pick a frontier node, finalise the node (its current parent becomes
 its final parent) and update information about the neighbours of the node.
 Dijkstra's algorithm uses a priority queue for the frontier nodes,
-ordered on the shortest distance to the node found so far.  At each
+ordered on the shortest distance from the start node
+to the node found so far.  At each
 stage the node with the minimum cost
 is removed for processing, and its neighbors have their information
 updated if a shorter path has now been found.

src/algorithms/explanations/PRIMExp.md

@@ -32,9 +32,9 @@ As these algorithms execute, we can classify nodes into three sets.
 These are:
 
 
-- "Finalised" nodes, for which the shortest or least costly path back to the start node has already
-been finalised (see below for further explanation of this),
-that is the final parent node has been determined and is recorded;
+- "Finalised" nodes, for which the path back to the start node has 
+been finalised, that is, the final parent node has been determined and recorded;
+
 
 - "Frontier" nodes, that are not finalised but are connected to a finalised node by a single edge; and
 
@@ -54,14 +54,6 @@ been finalised
 have their costs recomputed. Other algorithms use other data structures to keep track 
 of the frontier nodes.
 
-Saying the finalised nodes have "the least costly path" back to the start
-node may be a little misleading: Prim's algorithm uses the length of just
-*the first edge of the path* as the cost. At each step, Prim's algorithm
-chooses the node with the least *incremental* cost increase for the tree
-being constructed. Throughout the execution, if finalised has size *N*,
-it contains the smallest tree with *N* nodes that includes the start node.
-The paths from each node back to the start are generally not the shortest.
-
 In the presentation here, we do not give details of how the priority
 queue is implemented, but just emphasise it is a collection of nodes
 with associated costs and the node with the minimum cost is selected each