## Finding the shortest path – A * pathfinding

Two months ago I posted some interesting stuff I found: Shortest path. Let me explain, someone created a workbook that calculated the shortest path between a start cell and an end cell in a maze, and it did that using only formulas. Pretty amazing. The formulas were made from Dijkstra's Algorithm. I tried to create a larger maze but the workbook grew too large.

I then found the A * pathfinding algorithm and it is a lot easier for the computer to calculate. There is a great explanation of the A * pathfinding algorithm here: Introduction to A* Pathfinding.

Basically, there is an open list, a closed list and a final path. The open list contains cells that are being considered to find the final path. The closed list contains cells that we don't need to consider again.

They all contain coordinates or cells based on criteria. The criteria are:

- The distance to the END cell. (The distance to the END cell is calculated with the Manhattan distance method.
- The distance traveled from the start cell.

If the END cell is added to the open list, the closed list is finished calculating. Now it is time to find the final path. The final path is also found using the Manhattan distance method but it can only travel on cells in the closed list.

Here is an animated picture, it shows you how the macro works, simplified.

- The blue cell is the start cell
- The red cell is the end cell
- Gray cells are cells in the closed list.
- Green cells are the final path.
- Black cells are walls.

Here is a slightly more advanced "map".

This is a maze. I have removed the closed list cells to make the macro quicker.

### VBA Code

'Name macro Sub FindShortestPath1() 'Disable screen refresh Application.ScreenUpdating = False 'Dimension variables and their data types Dim G() As Variant Dim H() As Variant Dim N() As Variant Dim OL() As Variant Dim CL() As Variant Dim S() As Variant Dim E() As Variant Dim W() As Variant Dim Gv() As Variant Dim i As Single 'Redimension variables ReDim S(0 To 1) ReDim E(0 To 1) ReDim W(0 To 3, 0 To 1) ReDim OL(0 To 1, 0) ReDim CL(0 To 1, 0) ReDim Gv(0 To 3) 'Save values in cell range specified in named range "Area" to variable Rng Rng = Range("Area").Value 'Save the upper limit of rows from Rng variable to variable a a = UBound(Rng, 1) - 1 'Save the upper limit of columns from Rng variable to variable a b = UBound(Rng, 2) - 1 'Redimension variables G, H and N based on variable a and b ReDim G(0 To a, 0 To b) ReDim H(0 To a, 0 To b) ReDim N(0 To a, 0 To b) 'For ... next loop from 1 to the number of rows in Rng variable For R = 1 To UBound(Rng, 1) 'For ... next loop from 1 to the number of columns in Rng variable For C = 1 To UBound(Rng, 2) 'Check if cell Rng(r, C) is equal to "S" (start point) If Rng(R, C) = "S" Then 'Save number in variable R - 1 to variable S position 0 (zero) S(0) = R - 1 'Save number in variable C - 1 to variable S position 1 S(1) = C - 1 End If 'Check if cell Rng(r, C) is equal to "E" (end point) If Rng(R, C) = "E" Then E(0) = R - 1 E(1) = C - 1 End If 'Check if S(0) is not equal to "" and E(0) is not equal to "" If S(0) <> "" And E(0) <> "" Then Exit For Next C Next R 'Save number stored in S(0) to array variable CL (closed list) position 0,0 CL(0, 0) = S(0) 'row 'Save number stored in S(1) to array variable CL (closed list) position 1,0 CL(1, 0) = S(1) 'column 'Save -1 to array variable W position 0,0 W(0, 0) = -1 'Save 1 to array variable W position 1,0 W(1, 0) = 1 'Save 0 to array variable W position 2,0 W(2, 0) = 0 'Save 0 to array variable W position 3,0 W(3, 0) = 0 'Save 0 to array variable W position 0,1 W(0, 1) = 0 'Save 0 to array variable W position 1,1 W(1, 1) = 0 'Save -1 to array variable W position 2,1 W(2, 1) = -1 'Save 1 to array variable W position 3,1 W(3, 1) = 1 'Save boolean value False to variable Echk Echk = False 'Keep iterating until Echk is True Do Until Echk = True 'For ... next statement For i = 0 To UBound(CL, 2) 'For ... next statement For j = 0 To 3 'Save boolean value False to variable Echk chk = False 'Add number in CL position 0, i and W position j, 0 and save total to tr tr = CL(0, i) + W(j, 0) 'Add number in CL and W and save total to tc tc = CL(1, i) + W(j, 1) If tr < 0 Or tc < 0 Or tr > UBound(Rng, 1) Or tc > UBound(Rng, 2) Then chk = True Else For k = UBound(CL, 2) To 0 Step -1 If tr = CL(0, k) And tc = CL(1, k) Then chk = True Exit For End If Next k If Rng(tr + 1, tc + 1) = 1 Then chk = True For k = UBound(OL, 2) To 0 Step -1 If tr = OL(0, k) And tc = OL(1, k) Then chk = True If G(CL(0, i), CL(1, i)) + 1 < G(tr, tc) Then G(tr, tc) = G(CL(0, i), CL(1, i)) + 1 H(tr, tc) = Abs(tr - E(0)) + Abs(tc - E(1)) N(tr, tc) = G(tr, tc) + H(tr, tc) End If Exit For End If Next k If chk = False Then OL(0, UBound(OL, 2)) = tr OL(1, UBound(OL, 2)) = tc ReDim Preserve OL(UBound(OL, 1), UBound(OL, 2) + 1) G(tr, tc) = G(CL(0, i), CL(1, i)) + 1 H(tr, tc) = Abs(tr - E(0)) + Abs(tc - E(1)) N(tr, tc) = G(tr, tc) + H(tr, tc) If Rng(tr + 1, tc + 1) = "E" Then Echk = True End If End If Next j Next i If Echk <> True Then For i = LBound(OL, 2) To UBound(OL, 2) If OL(0, i) <> "" Then Nchk = N(OL(0, i), OL(1, i)) Exit For End If Next i For i = LBound(OL, 2) To UBound(OL, 2) If OL(1, i) <> "" Then If N(OL(0, i), OL(1, i)) < Nchk And N(OL(0, i), OL(1, i)) <> "" Then Nchk = N(OL(0, i), OL(1, i)) End If End If Next i For i = LBound(OL, 2) To UBound(OL, 2) If OL(0, i) <> "" Then If N(OL(0, i), OL(1, i)) = Nchk Then ReDim Preserve CL(UBound(CL, 1), UBound(CL, 2) + 1) CL(0, UBound(CL, 2)) = OL(0, i) OL(0, i) = "" CL(1, UBound(CL, 2)) = OL(1, i) OL(1, i) = "" End If End If Next i End If Loop tr = E(0) tc = E(1) Schk = False Do Until Schk = True For i = UBound(CL, 2) To 0 Step -1 If CL(0, i) = (tr + 1) And CL(1, i) = tc _ And (Rng(tr + 2, tc + 1) <> "W" _ And Rng(tr + 2, tc + 1) <> "1") _ Then Gv(0) = G(tr + 1, tc) If CL(0, i) = tr And CL(1, i) = (tc + 1) _ And (Rng(tr + 1, tc + 2) <> "W" _ And Rng(tr + 1, tc + 2) <> "1") _ Then Gv(1) = G(tr, tc + 1) If CL(0, i) = (tr - 1) And CL(1, i) = tc _ And (Rng(tr, tc + 1) <> "W" _ And Rng(tr, tc + 1) <> "1") _ Then Gv(2) = G(tr - 1, tc) If CL(0, i) = tr And CL(1, i) = (tc - 1) _ And (Rng(tr + 1, tc) <> "W" _ And Rng(tr + 1, tc) <> "1") _ Then Gv(3) = G(tr, tc - 1) For j = 0 To 3 If Gv(j) <> "" Then Nf = Gv(j) Next j For j = 0 To 3 If Gv(j) < Nf And Gv(j) <> "" Then Nf = Gv(j) Next j Next i Application.ScreenUpdating = True Select Case Nf Case Gv(0) tr = tr + 1 Range("Area").Cells(tr + 1, tc + 1) = "W" Rng(tr + 1, tc + 1) = "W" Case Gv(1) tc = tc + 1 Range("Area").Cells(tr + 1, tc + 1) = "W" Rng(tr + 1, tc + 1) = "W" Case Gv(2) tr = tr - 1 Range("Area").Cells(tr + 1, tc + 1) = "W" Rng(tr + 1, tc + 1) = "W" Case Gv(3) tc = tc - 1 Range("Area").Cells(tr + 1, tc + 1) = "W" Rng(tr + 1, tc + 1) = "W" End Select If Rng(tr + 2, tc + 1) = "S" _ Or Rng(tr + 1, tc + 2) = "S" _ Or Rng(tr, tc + 1) = "S" _ Or Rng(tr + 1, tc) = "S" Then Schk = True Loop Application.ScreenUpdating = True End Sub

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[…] I guess this is a task for vba but that will be another post. [UPDATE] The follow up post is here: Finding the shortest path – A * pathfinding […]

impressive!

Torstein,

Thank you but the idea is not mine. The A * algorithm is used in many computer games.

It is a very interesting technique and I believe I can make my macro run a lot quicker with some minor changes.

[UPDATE]

A quicker A * pathfinding algorithm

[…] Finding the shortest path – A * pathfinding […]

Nice explanation :) Here are some other visualizations with extra info, helping to better understand A* (along with forkeable examples): https://thewalnut.io/visualizer/visualize/7/6/