Algerian sauce: contains egg.
I found the following list of ingredients:
Rapeseed oil, onions (contains sulfite), glucose-fructose syrup, tomato puree, vinegar, egg yolk, salt, modified starch, bell pepper, onion powder, wheat starch, spice extract, preservatives: potassium sorbate – sodium benzoate, acidifier: lactic acid, chili powder 0.1%, thickener: carob seed meal, natural capsicum flavor, antioxidant: calcium disodium EDTA.
Samurai sauce: contains egg.
The same site has a simple recipe:
50 g of sambal oelek (which can be replaced by harissa)
150 g mayonnaise
4 tbsp of oil
2 tbsp of lemon
In a bowl combine all the ingredients and enjoy it!
Barbecue sauce: it depends.
Barbecue sauce is a very common thing to find in many fast food places and other shops all over the world, and there are many different brands and varieties. Many of them are vegetarian and even vegan, but some recipes include anchovies. These two sites contain (non-exhaustive) lists of brands of barbecue sauce that are definitely vegan.
In fact, the answer could be "it depends" to any of the above. Even for a type of food that clearly doesn't require any non-vegetarian contents, some manufacturers may add, for example, flavourings or colourings whose provenance is unknown and potentially non-vegetarian, while others may make an effort to make sure their products are fully vegan whenever possible. Without knowing the exact brand or ingredients list of what you ate, the best we can say is "probably" or "very likely".
To be clear, Algerian sauce and samurai sauce are vegetarian in the sense that this word is normally used in western countries (i.e. no meat or fish products, but eggs and dairy products are OK), but they're typically not vegan and also do not fall into the category of vegetarianism which you use, i.e. not lacto-vegetarian.