Delphi中Dijkstra最短路径搜索的优化 [英] Optimisation of a Dijkstra Shortest Path Search in Delphi
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问题描述
我正在寻找建议，以加快在权重图上执行Dijkstra最短路径搜索的过程，权重图是一个N x N的方矩阵.水平顶点的权重称为H(垂直顶点的权重为V). /p>
一张图片值一千字:
(来源: free.fr )
当然，这是更大应用程序的一部分，但是我在这里提取了相关内容:
unit Unit1;
interface
uses
Windows, Messages, SysUtils, Variants, Classes, Graphics, Controls, Forms,
Dialogs, StdCtrls;
const
N = 200; //Working on a grid of N x N, here for a quick test, in practice, it's more 10000
type
TForm1 = class(TForm)
Button1: TButton;
procedure FormCreate(Sender: TObject);
procedure Button1Click(Sender: TObject);
end;
TNode = class
public
ID, //Number of the Node
origin, //From which Node did I came?
weight : integer; //The total weight of the path to Node ID
done : boolean; //Is the Node already explored?
constructor Create(myID, myOrigin, myweight: integer);
end;
var
Form1: TForm1;
implementation
var
H, V : array of integer;
{$R *.dfm}
constructor TNode.Create(myID, myOrigin, myweight: integer);
begin
ID:=MyID;
origin:=MyOrigin;
weight:=MyWeight;
end;
{------------------------------------------------------------------------------}
Function GetNodeFromID(ID: integer; NodeList: TList) : TNode; overload;
var
I: Integer;
Node: TNode;
begin
result:=nil;
for I := 0 to NodeList.count-1 do
begin
Node := NodeList[i];
if Node.ID=ID then
begin
result:=Node;
break;
end;
end;
end;
{------------------------------------------------------------------------------}
Function GetNodeOfMiniWeight(NodeList: TList) : TNode; overload;
var
I, min: Integer;
Node: TNode;
begin
result:=nil;
min :=maxint;
for I := 0 to NodeList.count-1 do
begin
Node := NodeList[i];
if Node.done then continue;
if Node.weight < min then
begin
result:=Node;
min := Node.weight;
end;
end;
end;
{------------------------------------------------------------------------------}
procedure SearchShortestPath(origin,arrival: integer);
var
NewWeight: integer;
NodeList : Tlist;
NodeFrom, //The Node currently being examined
NodeTo :TNode; //The Node where it is intented to go
s : string;
begin
NodeList := Tlist.Create;
NodeFrom := TNode.Create(origin,MaxInt,0);
NodeList.Add(NodeFrom);
while not (NodeFrom.ID = arrival) do //Arrived?
begin
//Path toward the top
if (NodeFrom.ID > N-1) //Already at the top of the grid
and not(NodeFrom.origin-NodeFrom.ID = N) then //Coming from the top
begin
NewWeight:=NodeFrom.weight + H[NodeFrom.ID-N];
NodeTo := GetNodeFromID(NodeFrom.ID-N, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID-N,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
//Path toward the bottom
if (NodeFrom.ID < N*N-N) //Already at the bottom of the grid
and not(NodeFrom.Origin- NodeFrom.ID = N) then //Coming from the bottom
begin
NewWeight:=NodeFrom.weight + H[NodeFrom.ID];
NodeTo := GetNodeFromID(NodeFrom.ID+N, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID+N,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
//Path toward the right
if not(NodeFrom.ID mod N = N-1) //Already at the extrem right of the grid
and not(NodeFrom.Origin - NodeFrom.ID = 1) then //Coming from the right
begin
NewWeight:=NodeFrom.weight + V[NodeFrom.ID];
NodeTo := GetNodeFromID(NodeFrom.ID+1, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID+1,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
//Path toward the left
if not (NodeFrom.ID mod N = 0) //Already at the extrem right of the grid
and not(NodeFrom.Origin - NodeFrom.ID = -1) then //Coming from the left
begin
NewWeight:=NodeFrom.weight + V[NodeFrom.ID-1];
NodeTo := GetNodeFromID(NodeFrom.ID-1, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID-1,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
NodeFrom.done :=true;
NodeFrom:=GetNodeOfMiniWeight(NodeList);
end;
s:='The shortest path from '
+ inttostr(arrival) + ' to '
+ inttostr(origin) + ' is : ';
//Get the path
while (NodeFrom.ID <> origin) do
begin
s:= s + inttostr(NodeFrom.ID) + ', ';
NodeFrom:=GetNodeFromID(NodeFrom.Origin, NodeList);
end;
s:= s + inttostr(NodeFrom.ID);
ShowMessage(s);
end;
procedure TForm1.Button1Click(Sender: TObject);
begin
SearchShortestPath(Random(N*N),Random(N*N));
end;
procedure TForm1.FormCreate(Sender: TObject);
var
I: Integer;
begin
//Initialisation
randomize;
SetLength(V,N*N);
SetLength(H,N*N);
for I := 0 to N*N-1 do
begin
V[I]:=random(100);
H[I]:=random(100);
end;
end;
end.
代码在例程GetNodeFromID和GetNodeOfMiniWeight中花费了大部分时间，并且在创建节点上花费了大量时间.
我认为我可以使用二进制搜索，但是由于它需要对列表进行排序，因此我认为我会浪费时间对列表进行排序.欢迎任何建议.
解决方案
我为稀疏图实现了Dijkstra最短路径算法的修改.您的图非常稀疏(E<< V ^ 2).此代码使用基于二进制堆的优先级队列，该队列包含(VerticeNum，DistanceFromSource)对作为TPoint，并按Distance(Point.Y)进行排序.它揭示了对数线性(接近线性)渐近行为.小图示例:
i5-4670时间
N V time, ms
100 10^4 ~15
200 4*10^4 ~50-60 //about 8000 for your implementation
400 1.6*10^5 100
1600 2.5*10^6 1300
6400 4*10^7 24000
10000 10^8 63000
//~max size in 32-bit OS due to H,V arrays memory consumption
代码:
function SparseDijkstra(Src, Dest: integer): string;
var
Dist, PredV: array of integer;
I, j, vert, CurDist, toVert, len: integer;
q: TBinaryHeap;
top: TPoint;
procedure CheckAndChange;
begin
if Dist[vert] + len < Dist[toVert] then begin
Dist[toVert] := Dist[vert] + len;
PredV[toVert] := vert;
q.Push(Point(toVert, Dist[toVert]));
//old pair is still stored but has bad (higher) distance value
end;
end;
begin
SetLength(Dist, N * N);
SetLength(PredV, N * N);
for I := 0 to N * N - 1 do
Dist[I] := maxint;
Dist[Src] := 0;
q := TBinaryHeap.Create(N * N);
q.Cmp := ComparePointsByY;
q.Push(Point(Src, 0));
while not q.isempty do begin
top := q.pop;
vert := top.X;
CurDist := top.Y;
if CurDist > Dist[vert] then
continue; //out-of-date pair (bad distance value)
if (vert mod N) <> 0 then begin // step left
toVert := vert - 1;
len := H[toVert];
CheckAndChange;
end;
if (vert div N) <> 0 then begin // step up
toVert := vert - N;
len := V[toVert];
CheckAndChange;
end;
if (vert mod N) <> N - 1 then begin // step right
toVert := vert + 1;
len := H[vert];
CheckAndChange;
end;
if (vert div N) <> N - 1 then begin // step down
toVert := vert + N;
len := V[vert];
CheckAndChange;
end;
end;
q.Free;
// calculated data may be used with miltiple destination points
result := '';
vert := Dest;
while vert <> Src do begin
result := Format(', %d', [vert]) + result;
vert := PredV[vert];
end;
result := Format('%d', [vert]) + result;
end;
procedure TForm1.Button2Click(Sender: TObject);
var
t: Dword;
I, row, col: integer;
begin
t := GetTickCount;
if N < 6 then // visual checker
for I := 0 to N * N - 1 do begin
col := I mod N;
row := I div N;
Canvas.Font.Color := clBlack;
Canvas.Font.Style := [fsBold];
Canvas.TextOut(20 + col * 70, row * 70, inttostr(I));
Canvas.Font.Style := [];
Canvas.Font.Color := clRed;
if col < N - 1 then
Canvas.TextOut(20 + col * 70 + 30, row * 70, inttostr(H[I]));
Canvas.Font.Color := clBlue;
if row < N - 1 then
Canvas.TextOut(20 + col * 70, row * 70 + 30, inttostr(V[I]));
end;
Memo1.Lines.Add(SparseDijkstra({0, n*n-1}random(N * N), random(N * N)));
Memo1.Lines.Add('time ' + inttostr(GetTickCount - t));
end;
TQPriorityQueue是内部使用的类，但是您可以尝试任何基于堆的优先级队列的实现.例如，此.您必须在此模块中将Pointer更改为TPoint，将Word更改为Integer.
Edit2: 我已经用BinaryHeap方法替换了过程中的专有队列方法名称.
I'm looking for advices to speed up my implementation of Dijkstra Shortest Path Search on a weighted graph which is a square matrix N x N. The weight on horizontal vertice is called H (resp. V on vertical ones).
A picture is worth a thousand words:
(source: free.fr)
Of course, this is part of a bigger application, but I've extracted the relevant bit here:
unit Unit1;
interface
uses
Windows, Messages, SysUtils, Variants, Classes, Graphics, Controls, Forms,
Dialogs, StdCtrls;
const
N = 200; //Working on a grid of N x N, here for a quick test, in practice, it's more 10000
type
TForm1 = class(TForm)
Button1: TButton;
procedure FormCreate(Sender: TObject);
procedure Button1Click(Sender: TObject);
end;
TNode = class
public
ID, //Number of the Node
origin, //From which Node did I came?
weight : integer; //The total weight of the path to Node ID
done : boolean; //Is the Node already explored?
constructor Create(myID, myOrigin, myweight: integer);
end;
var
Form1: TForm1;
implementation
var
H, V : array of integer;
{$R *.dfm}
constructor TNode.Create(myID, myOrigin, myweight: integer);
begin
ID:=MyID;
origin:=MyOrigin;
weight:=MyWeight;
end;
{------------------------------------------------------------------------------}
Function GetNodeFromID(ID: integer; NodeList: TList) : TNode; overload;
var
I: Integer;
Node: TNode;
begin
result:=nil;
for I := 0 to NodeList.count-1 do
begin
Node := NodeList[i];
if Node.ID=ID then
begin
result:=Node;
break;
end;
end;
end;
{------------------------------------------------------------------------------}
Function GetNodeOfMiniWeight(NodeList: TList) : TNode; overload;
var
I, min: Integer;
Node: TNode;
begin
result:=nil;
min :=maxint;
for I := 0 to NodeList.count-1 do
begin
Node := NodeList[i];
if Node.done then continue;
if Node.weight < min then
begin
result:=Node;
min := Node.weight;
end;
end;
end;
{------------------------------------------------------------------------------}
procedure SearchShortestPath(origin,arrival: integer);
var
NewWeight: integer;
NodeList : Tlist;
NodeFrom, //The Node currently being examined
NodeTo :TNode; //The Node where it is intented to go
s : string;
begin
NodeList := Tlist.Create;
NodeFrom := TNode.Create(origin,MaxInt,0);
NodeList.Add(NodeFrom);
while not (NodeFrom.ID = arrival) do //Arrived?
begin
//Path toward the top
if (NodeFrom.ID > N-1) //Already at the top of the grid
and not(NodeFrom.origin-NodeFrom.ID = N) then //Coming from the top
begin
NewWeight:=NodeFrom.weight + H[NodeFrom.ID-N];
NodeTo := GetNodeFromID(NodeFrom.ID-N, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID-N,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
//Path toward the bottom
if (NodeFrom.ID < N*N-N) //Already at the bottom of the grid
and not(NodeFrom.Origin- NodeFrom.ID = N) then //Coming from the bottom
begin
NewWeight:=NodeFrom.weight + H[NodeFrom.ID];
NodeTo := GetNodeFromID(NodeFrom.ID+N, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID+N,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
//Path toward the right
if not(NodeFrom.ID mod N = N-1) //Already at the extrem right of the grid
and not(NodeFrom.Origin - NodeFrom.ID = 1) then //Coming from the right
begin
NewWeight:=NodeFrom.weight + V[NodeFrom.ID];
NodeTo := GetNodeFromID(NodeFrom.ID+1, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID+1,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
//Path toward the left
if not (NodeFrom.ID mod N = 0) //Already at the extrem right of the grid
and not(NodeFrom.Origin - NodeFrom.ID = -1) then //Coming from the left
begin
NewWeight:=NodeFrom.weight + V[NodeFrom.ID-1];
NodeTo := GetNodeFromID(NodeFrom.ID-1, NodeList);
if NodeTo <> nil then
begin
if NodeTo.weight > NewWeight then
begin
NodeTo.Origin:=NodeFrom.ID;
NodeTo.weight:=NewWeight;
end;
end
else
begin
NodeTo := TNode.Create(NodeFrom.ID-1,NodeFrom.ID,NewWeight);
NodeList.Add(NodeTo);
end;
end;
NodeFrom.done :=true;
NodeFrom:=GetNodeOfMiniWeight(NodeList);
end;
s:='The shortest path from '
+ inttostr(arrival) + ' to '
+ inttostr(origin) + ' is : ';
//Get the path
while (NodeFrom.ID <> origin) do
begin
s:= s + inttostr(NodeFrom.ID) + ', ';
NodeFrom:=GetNodeFromID(NodeFrom.Origin, NodeList);
end;
s:= s + inttostr(NodeFrom.ID);
ShowMessage(s);
end;
procedure TForm1.Button1Click(Sender: TObject);
begin
SearchShortestPath(Random(N*N),Random(N*N));
end;
procedure TForm1.FormCreate(Sender: TObject);
var
I: Integer;
begin
//Initialisation
randomize;
SetLength(V,N*N);
SetLength(H,N*N);
for I := 0 to N*N-1 do
begin
V[I]:=random(100);
H[I]:=random(100);
end;
end;
end.
The code spend most of the time in the routines: GetNodeFromID and GetNodeOfMiniWeight, and a substantial time to create nodes.
I thought that I could use a binary search, but since it requires the list to be sorted, I think that I'll loose the time in sorting the list. Any advice is welcome.
解决方案
I've implemented modification of Dijkstra Shortest Path algorithm for sparsed graphs. Your graph is very sparsed (E << V^2). This code uses priority queue based on binary heap, that contains (VerticeNum, DistanceFromSource) pairs as TPoints, ordered by Distance (Point.Y). It reveals loglinear (close to linear) asymptotic behavior. Example for small graph:
Times for i5-4670
N V time, ms
100 10^4 ~15
200 4*10^4 ~50-60 //about 8000 for your implementation
400 1.6*10^5 100
1600 2.5*10^6 1300
6400 4*10^7 24000
10000 10^8 63000
//~max size in 32-bit OS due to H,V arrays memory consumption
code:
function SparseDijkstra(Src, Dest: integer): string;
var
Dist, PredV: array of integer;
I, j, vert, CurDist, toVert, len: integer;
q: TBinaryHeap;
top: TPoint;
procedure CheckAndChange;
begin
if Dist[vert] + len < Dist[toVert] then begin
Dist[toVert] := Dist[vert] + len;
PredV[toVert] := vert;
q.Push(Point(toVert, Dist[toVert]));
//old pair is still stored but has bad (higher) distance value
end;
end;
begin
SetLength(Dist, N * N);
SetLength(PredV, N * N);
for I := 0 to N * N - 1 do
Dist[I] := maxint;
Dist[Src] := 0;
q := TBinaryHeap.Create(N * N);
q.Cmp := ComparePointsByY;
q.Push(Point(Src, 0));
while not q.isempty do begin
top := q.pop;
vert := top.X;
CurDist := top.Y;
if CurDist > Dist[vert] then
continue; //out-of-date pair (bad distance value)
if (vert mod N) <> 0 then begin // step left
toVert := vert - 1;
len := H[toVert];
CheckAndChange;
end;
if (vert div N) <> 0 then begin // step up
toVert := vert - N;
len := V[toVert];
CheckAndChange;
end;
if (vert mod N) <> N - 1 then begin // step right
toVert := vert + 1;
len := H[vert];
CheckAndChange;
end;
if (vert div N) <> N - 1 then begin // step down
toVert := vert + N;
len := V[vert];
CheckAndChange;
end;
end;
q.Free;
// calculated data may be used with miltiple destination points
result := '';
vert := Dest;
while vert <> Src do begin
result := Format(', %d', [vert]) + result;
vert := PredV[vert];
end;
result := Format('%d', [vert]) + result;
end;
procedure TForm1.Button2Click(Sender: TObject);
var
t: Dword;
I, row, col: integer;
begin
t := GetTickCount;
if N < 6 then // visual checker
for I := 0 to N * N - 1 do begin
col := I mod N;
row := I div N;
Canvas.Font.Color := clBlack;
Canvas.Font.Style := [fsBold];
Canvas.TextOut(20 + col * 70, row * 70, inttostr(I));
Canvas.Font.Style := [];
Canvas.Font.Color := clRed;
if col < N - 1 then
Canvas.TextOut(20 + col * 70 + 30, row * 70, inttostr(H[I]));
Canvas.Font.Color := clBlue;
if row < N - 1 then
Canvas.TextOut(20 + col * 70, row * 70 + 30, inttostr(V[I]));
end;
Memo1.Lines.Add(SparseDijkstra({0, n*n-1}random(N * N), random(N * N)));
Memo1.Lines.Add('time ' + inttostr(GetTickCount - t));
end;
Edit: TQPriorityQueue is class for internal use, but you can try any implementation of heap-based priority queue. For example, this one. You have to change Pointer to TPoint, Word to Integer in this module.
Edit2: I've replaced proprietary queue method names in my procedure by BinaryHeap methods.
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