Just addressing the triangle inequality violations, hopefully the extra factor they're optimising for is common sense. You don't necessarily want the shortest or fastest route, as it can lead to chaos and destruction. If you want your directions to prefer the major routes that are truck-friendly and can cope with having every sat-nav-following driver sent down them, you quickly discard the triangle inequality[1].

If Y is a narrow residential street between X and Z, you probably do only want to use the shortcut via Y if the user explicitly asks for X-Y-Z. If they ask for X-Z, they should stick to major roads even if it's a bit further and takes a bit longer. It's similar to Braess's paradox - if everyone tries to take the shortest, fastest route, the resulting congestion means that it's not the fastest route for anyone any more. From here we stray from graph theory into game theory.

[1] In fact, any hope that the distances produced will be a distance function in the mathematical sense dies when you allow one-way roads and lose the symmetry requirement. Losing the triangle inequality too is just rubbing salt in the wound.

Just addressing the triangle inequality violations, hopefully the extra factor they're optimising for is common sense. You don't necessarily want the shortest or fastest route, as it can lead to chaos and destruction. If you want your directions to prefer the major routes that are truck-friendly and can cope with having every sat-nav-following driver sent down them, you quickly discard the triangle inequality[1].

If Y is a narrow residential street between X and Z, you probably do only want to use the shortcut via Y if the user explicitly asks for X-Y-Z. If they ask for X-Z, they should stick to major roads even if it's a bit further and takes a bit longer. It's similar to Braess's paradox - if everyone tries to take the shortest, fastest route, the resulting congestion means that it's not the fastest route for anyone any more. From here we stray from graph theory into game theory.

[1] In fact, any hope that the distances produced will be a distance function in the mathematical sense dies when you allow one-way roads and lose the symmetry requirement. Losing the triangle inequality too is just rubbing salt in the wound.

Just addressing the triangle inequality violations, hopefully the extra factor they're optimising for is common sense. You don't necessarily want the shortest or fastest route, as it can lead to chaos and destruction. If you want your directions to prefer the major routes that are truck-friendly and can cope with having every sat-nav-following driver sent down them, you quickly discard the triangle inequality[1].

If Y is a narrow residential street between X and Z, you probably do only want to use the shortcut via Y if the user explicitly asks for X-Y-Z. If they ask for X-Z, they should stick to major roads even if it's a bit further and takes a bit longer. It's similar to Braess's paradox - if everyone tries to take the shortest, fastest route, the resulting congestion means that it's not the fastest route for anyone any more. From here we stray from graph theory into game theory.

[1] In fact, any hope that the distances produced will be a distance function in the mathematical sense dies when you allow one-way roads and lose the symmetry requirement. Losing the triangle inequality too is just rubbing salt in the wound.

Just addressing the triangle inequality violations, hopefully the extra factor they're optimising for is common sense. You don't necessarily want the shortest or fastest route, as it can lead to chaos and destruction. If you want your directions to prefer the major routes that are truck-friendly and can cope with having every sat-nav-following driver sent down them, you quickly discard the triangle inequality[1].

If Y is a narrow residential street between X and Z, you probably do only want to use the shortcut via Y if the user explicitly asks for X-Y-Z. If they ask for X-Z, they should stick to major roads even if it's a bit further and takes a bit longer. It's similar to Braess's paradox - if everyone tries to take the shortest, fastest route, the resulting congestion means that it's not the fastest route for anyone any more. From here we stray from graph theory into game theory.

[1] In fact, any hope that the distances produced will be a distance function in the mathematical sense dies when you allow one-way roads and lose the symmetry requirement. Losing the triangle inequality too is just rubbing salt in the wound.